how do we snuff out or obscure the light and heat of our Sun?

You cannot completely hide the energy emitted by the Sun.

You can just shift it to longer wavelengths as a consequence of using the usable content of the emitted energy, but, as madam Thermodynamic states, any transition results in some form of heat being emitted by a system at a lower temperature. Once you reach 0 K, you cannot extract any more usable work, and you have reached the thermal death.

I.e. the visible light we get on Earth and warms up a car parked under the Sun comes from the surface of the Sun being at about 6000 K, while the car re-emits it in the infrared, at longer wavelength.

Even if you were able to convert the emitted energy into mass, you would have the problem of dissipating all the gravitational energy of the resulting mass plus the Sun, else the whole thing would ignite again. And dissipating that energy would give an energy emission.

Problem: even if you could just stick a blanket over the sun, it is probably already too late. The solar system formed more than 4 billion years ago, and for all that time anyone who was watching and had suitably acute vision would have been able to see Sol, and almost certainly the protoplanetary disc around it and later the planets themselves. Certainly, anyone or anything with K2-level technology will know where the sun is, and have a pretty good guess at what sort of planetary system it harbours.

Stars don't just poof out of existence without a trace. Even being eaten by a black hole is a pretty drawn-out and violent event. Those planet eaters? They'll see, and they'll know. You can't pull the wool over their eyes. They'll see that little yellow dot fade away, the little yellow dot that is making unscheduled departure from the main sequence and clearly isn't behaving like a natural star should. They'll come and take a look at the clear and unambiguous evidence of intelligent agency, because intelligence generally comes from planetary systems and that means more food.

So give up on your plan. You gonna get ate.

Instead, read up on mechanisms of planetary relocation, preferably with reactionless drives, and boost the Earth out of Sol's gravity well as soon as you possibly can. They might notice the absense of the earth, but given the rest of the solar system to snack upon, and assuming suitable stealthiness of the fleeing world, maybe, just maybe, they won't find you.

Good luck, cos you're gonna need it.

As an aside, given that you have the potential power to snuff out a sun, you will certainly have the power to do simpler things. I'd see about building a giant "blackhouse" roof covering the earth (or as much of it as seemed practical), in a technique sometimes called paraterraforming or a worldhouse. Life might not able to be as thermodynamically exciting as it used to be, absent energy input from the sun, but you've got access to a lot of fusion fuel in the seas and you can build and run a lot of grow lights with that.

Probably would be insufficiently gritty for your narrative needs, but, y'know, the category of "things easier to do than surreptitiously turn off an entire main sequence star" includes a lot of stuff like this.

I'm looking for an alternative to building a complete Dyson sphere because my goal as an author is to write stories about the dark cold earth.

There isn't sufficient matter in the solar system to build a Dyson sphere or swarm at one astronomical unit or farther (i.e. with the Earth inside the sphere) more than four meters thick. So you'd have to build it much closer to the sun, perhaps inside Mercury's orbit, to get it thick enough to obscure the Sun. That would still give you your dark cold Earth (presuming you didn't take apart the Earth to make the Dyson sphere).

Surface area is proportional to the square of the radius. So a quarter of the radius (which is about where Mercury is) would give sixteen times the thickness. I'm not sure how much farther in you can make the Dyson sphere before it would get too hot.

You may also find that it makes more sense to build two spheres. The inner one catches the light but reradiates a significant portion. The outer one catches that light and reradiates it with a different wavelength as already suggested. In between the spheres might be the gaseous portion of the mass, soaking up some of the energy output.

A star shines because it has mass...

You put enough mass together, it gets a dense core, heats up, and voila, solar fusion. Yes, that's an oversimplification, but, fundamentally, making a star not shine would require removing its mass.

So let's figure out how to remove mass...

But you said you want the mass to stay at the center so that orbital mechanics doesn't shift for the planets. That suggests a delicate surgery -- divide the sun into N parts where each part is less than the minimum stellar mass, and let the masses orbit around the original gravitational center of the sun (because gravity works as if all of an object's or system's mass were at the center of gravity for everything in orbit around it). The smallest theoretical mass for a star with same metallicity of Sol to support nuclear fusion is 75 x mass of Jupiter. Sol is ~1000x mass of Jupiter. So we'll need to cut Sol into 14 chunks.

14 is an awkward number, so let's make it 20, and make each one drift away so that we get an icosahedron of chunks. A, that'll make it easier for our imaginary thrust system to arrange things geometrically, and B, that'll make a great book jacket cover. It might also keep the star's mass well-balanced, so, again, we don't have to worry about the planets feeling the effect. I did mention this was a delicate surgery, right? Yes? Good. Moving on...

Having divided the star, the core should cool down, fusion stops: Good night, starshine! (Gonna need a new musical number when future humans perform the stage play Hair.)

You want your Giant Space Scissors to push the chunks apart at just the right speed so that they eventually drift back together when the threat is passed.

Design of Giant Space Scissors

You have a problem of scale. Stars are BIG. Really big. Like hurling-Earth-into-Sol-won't-break-it big. That means we are NOT talking about any sort of mechanical scissors.

You're going to need a chemistry solution -- something that you can seed into 20 sides of Sol that starts some sort of repulsive reaction. I think you're going to invent handwavium or unobtanium for this. Here's my attempt...

Each seed must be a little thing that will pull Sol's mass to itself and overcome the gravity that is holding Sol together. As it does this, it needs to generate either positive or negative charge such that it pushes away from the seeds around it. ALAS -- there is no way to color an icosahedron with only two colors such that no two adjacent edges have the same color. You can only do that with an octagon. So you can't just rely on alternating positive and negative electrical charge to do the push back (i.e., have some "positive seeds" and some "negative seeds"). So, more handwaving: we'll assume that the act of the seed pulling in mass somehow imparts momentum to the chunks along the vector from which they are pulling the most mass (i.e. away from star's core). That's nice because it means that momentum will eventually be overcome by gravity and the chunks will drift back together.

Note: Having the seeds become uniformly positively (or negatively) charged would cause them to push away from each other, but if that pushback was strong enough to overcome gravity, they'd never drift back together. The momentum solution is better in my opinion.

Problems:

1. It'll require a lot of set up to get the seeds arranged around the sun. We're talking years of construction and travel, not weeks.


2. The process of dividing the sun won't be fast. Sure, you can posit a geometric expansion of speed as the seed grows and acquires mass, but we are still talking SOLAR MASS. It takes a while for that much mass to move, even if the impulse is given atom by atom in some sort of known-physics-defying tractor beam that the seed is emitting.


3. It'll take a while for the core chunks to cool. Possibly as long as tens or hundreds of thousands of years. Not a lot of ways to accelerate the cooling. (Thanks to user @MikeScott for the link.)

Basically, these space bugs better be a long way away. Centuries. Millennia. Long enough for humans to have developed multiple spacefaring civilizations, collapsed back to bronze age, and rebuilt again. So you may want to posit FTL travel because FTL MUST grant time travel, so humanity can launch the probes toward the sun and backward in time. Just be aware that once you allow for violations of causality, forever will it dominate your destiny. (Yoda lives in a galaxy that has FTL, so he knows this problem. Or will know it. Hard to say with relativity + FTL.)

Or you could NOT violate known physics and just put the whole of humanity on a space ship that gets up close to speed of light (C) and then let relativity and time dilation do the work for you, so humanity is only gone a few weeks and comes back to a much changed Earth, but no space bugs. You get to solve the acceleration and deceleration problems yourself if you choose this solution!

4. Something has to tell the seeds to shut off. When the chunks drift back together, stellar fusion won't ignite if the seeds keep the chunks isolated enough that the "swiss cheese" of the chunk spheres provide enough venting for the heat. Ok... let's just assume that there isn't enough venting for the heat, stellar fusion reignites, and the seeds get pulled toward the core where it eventually becomes too hot and they lose whatever tractor beam powers they had. Ok, so not a problem. Nothing is a problem with enough handwavium. :-)


5. That star will be cranky when it comes back online. Expect a lot of solar flares while it settles back into its mainline again. That may cook the Earth, but you'll be used to living underground by then, so just add a few thousand years to the underground time. Unless you pick that time dilation solution. Or you could have everyone upload their minds into a computer or... yeah, we've all read sci-fi... pick your solution here.

Seriously... we are talking STARS here. Anything you do involving space blows human lifespans out of the water. You're going to need to deal with that in any story that has even a passing acquaintance with science. That's the saddest part of modern physics: The Stars Are Not For Humanity. (Thank you, Arthur C. Clarke, for summing up the crushing of geek dreams.)

PS: Book title: "Divisions of the Sol" -- the dividing of the sun set against the divisions in a young protagonist's heart as he/she longs for a partner against family wishes. It'll sell like hotcakes to the folks who like their sci-fi soft and their romance hard. :-)

Basic Requirements

We are talking about disassembling a star here. While this is certainly possible within known physics, it is something that requires the infrastructure, resources, and energy of a K 2.something civilization. Therein lies the issue. Even with all the knowledge and technology, is disassembling the sun really the smartest option? You gonna have to explain why other options weren't chosen. Running away seems quite easy in comparison.

Shkadov Thruster

This is basically using half a Dyson sphere to accelerate the sun. The issue is that the acceleration is abysmal. This video and the Wikipedia article on the subject might be interesting.

For a star such as the Sun, with luminosity 3.85 × 1026 W and mass 1.99 × 1030 kg, the total thrust produced by reflecting half of the solar output would be 1.28 × 1018 N. After a period of one million years this would yield an imparted speed of 20 m/s, with a displacement from the original position of 0.03 light-years. After one billion years, the speed would be 20 km/s and the displacement 34,000 light-years, a little over a third of the estimated width of the Milky Way galaxy.

Fleeing with Earth

There are several ways to do this. This video discusses it in detail. While the thrust you can apply to Earth before the engines will push the continent they are on into the mantle and deform the Earth is still small, it isn´t as abysmal as the Shkadov thruster idea. Fuel will be a problem, but putting several layers of shielding and hydrogen for fuel and reaction mass around Earth will give you a dark planet theme. Putting this armored Earth in orbit around Jupiter and using fusion candles, which are discussed in the linked post. The following description is from said post.

Build a fusion candle. It's called a "candle" because you're going to burn it at both ends. The center section houses a set of intakes that slurp up gas giant atmosphere and funnel it to the fusion reactors at each end. Shove one end deep down inside the gas giant, and light it up. It keeps the candle aloft, hovering on a pillar of flame. Light up the other end, which now spits thrusting fire to the sky. Steer with small lateral thrusters that move the candle from one place to another on the gas giant. Steer very carefully, and signal your turns well in advance. This is a big vehicle. Balance your thrusting ends with exactness. You don't want to crash your candle into the core of the giant, or send it careening off into a burningly elliptical orbit. When the giant leaves your system, it will take its moons with it. This is gravity working for you. Put your colonists on the moons.For safety's sake, the moons should orbit perpendicular to the direction of travel. Otherwise your candle burns them up.They should also rotate in the same plane, with one pole always illuminated by your candle (think "portable sunlight") The other pole absorbing the impact of whatever interstellar debris you should hit (think "don't build houses on this side")

Colonial Fleet

Just get everyone onto colony or world-ships and run for it. You wanna take the Earth with you? That's easy. Just peel it like an orange and place the crust fragments in rotating habitats on said vessels. Sounds crazy? Keep in mind that you just proposed to disassemble the sun.

Peeling earth is relatively easy given the knowledge humanity was given. There will be a ton of very advanced geo-engineering knowledge in there. Dig extensive tunnel systems under the continental crust, place engines there and keep them thrusting until the huge, domed over parts of Earth sit in prepared spin habitats.

I´ll calculate in a moment why I believe that fleeing is the superior choice. Several of the options will still give you decent dark Earth setting.

The Logistics

Whatever you attempt the only way to do it in any reasonable time frame are self replicating machines. Say goodbye to Mercury, Mars, the asteroid belt and probably more of the planets, as they´ll be needed for construction materials. Beyond that, you´ll need a lot of time. With all of that out of the way.

Starlifting

Starlifting is discussed in detail in this video. The issue with starlifting is that it takes a long time. A very long time. Using 100% of the energy output of the sun will allow you to remove 0.000003% of the Sun's total mass per year. After only 334 million years the sun will have been disassembled. (it will be even longer if you intend to collect the material for later use) The swarm would have to be out past the Andromeda galaxy assuming they can move with a speed near light speed. Fortifying the entire Milky Way galaxy and turning every star into a Nicoll-Dyson Laser (the satellites of a Dyson swarm act as a phased array laser emitter capable of delivering their energy to a planet-sized target at a range of millions of light years) to fry the bastards out of the sky seems easier and more practical. That said, there are three main methods suggested in the Wikipedia article:

Thermal-driven outflow

The simplest system for star lifting would increase the rate of solar wind outflow by directly heating small regions of the star's atmosphere. This would produce a large and sustained eruption similar to a solar flare at the target location, feeding the solar wind. The resulting outflow would be collected by using a ring current around the star's equator to generate a powerful toroidal magnetic field with its dipoles over the star's rotational poles. This would deflect the star's solar wind into a pair of jets aligned along its rotational axis passing through a pair of magnetic rocket nozzles. The magnetic nozzles would convert some of the plasma's thermal energy into outward velocity, helping cool the outflow.

"Huff-n-Puff"

In this system the ring of particle accelerators would not be in orbit, instead depending on the outward force of the magnetic field itself for support against the star's gravity. To inject energy into the star's atmosphere the ring current would first be temporarily shut down, allowing the particle accelerator stations to begin falling freely toward the star's surface. Once the stations had developed sufficient inward velocity the ring current would be reactivated and the resulting magnetic field would be used to reverse the stations' fall. This would "squeeze" the star, propelling stellar atmosphere through the polar magnetic nozzles. The ring current would be shut down again before the ring stations achieved enough outward velocity to throw them too far away from the star, and the star's gravity would be allowed to pull them back inward to repeat the cycle. A single set of ring stations would result in a very intermittent flow. It is possible to smooth this flow out by using multiple sets of ring stations, with each set operating in a different stage of the Huff-n-Puff cycle at any given moment so that there is always one ring "squeezing". This would also smooth out the power requirements of the system over time.

Centrifugal acceleration

The two magnetic nozzles would then be located on the star's equator. To increase the rate of outflow through these two equatorial jets, the ring system would be rotated around the star at a rate significantly faster than the star's natural rotation. This would cause the stellar atmosphere swept up by the magnetic field to be flung outward. This method suffers from a number of significant complications compared to the others. Rotating the ring in this manner would require the ring stations to use powerful rocket thrust, requiring both large rocket systems and a large amount of reaction mass.

Deathsinger

This method is a bit more speculative than the others. Science-fiction author Alastair Reynolds proposed to use a gravity laser (Gwaser, Gaser, Graser, or Glaser) to "sing" a hole down to the core of a star. This would create a beam of stellar material powered by the internal pressure of the star, which would slowly deplete the star. In the novel Redemption Ark this happens on a timeline of a few years. This might be the option that would fit your purposes best.