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Due to various minor conflicts my manager has asked me to resign. Its a situation in which I'm really not a great fit for the team rather than any specific cause for termination. Because of that he gave me a few months to look for another job. Now I'm coming up with two weeks left on the job and I'm ready to let my team know that I'm leaving. But I haven't lined another job up yet. So I'm unsure how to announce my resignation without indicating that I've been terminated and in a way that will keep coworkers from asking about my next gig (since I don't have one). I want to keep the level of awkward at a bare minimum.
In the end, I have already handed in my resignation, so the topic of whether to resign or wait until they fire me is moot.

My question is therefore, what are some great ways to communicate to coworkers that I’m moving on without indicating it was an involuntary termination and that I’ve yet to secure another job?

My goal is to reduce awkwardness as much as possible.

Good afternoon Team,

I wanted to let you know first that my last day at [business] will be on the [day]. I sincerely appreciate having had the opportunity to work with you all. I’ve learned so much from each of you. Even so I am looking forward to the new challenges my next position will bring. [My Boss] will be making the announcement to the full department tomorrow. I’ll be sure to send out my final farewell later later on with my contact information and linked-in profile.

Best Wishes,

[My Name]

When they ask, I'm considering saying something like "Ill let everyone know [where I'm going] in my final farewell letter". Hopefully I'll actually have an an offer by then.

Based on your comment, you are located in the United States.

If I were you, I would not resign. I would continue doing my job and give the best performance I could (Edit: Exactly for getting unemployment benefits - See first comment), then search and secure a job ASAP. Until they decide to fire me.

At this point, do NOT do what they are asking you to do. And once they fire you leave and take your experience and performance review/records with you, apply for unemployment benefits and get paid while finding the next job.

Now for the team, I would not pull my issues with my manager into the team. Don't tell the team anything and if you get fired, leave this worry behind and let the management figure out what to do and how to tell the team.

If for any reason you have to tell your team, I would just send an email. Something like this would work:

Team,

I have been asked to leave and effective last day is xx/xx/xxxx. It
was my pleasure meeting and working with each one of you. Please let
me know if you have any questions and I will be more than happy to help before I leave.

Since you don't have a new job lined up, I would be honest with your team, and say that you didn't feel you were a good fit, so you and the manager agreed to part ways. I would emphasize that this was a mutual decision, and you've thought about it for a while. The point to all this is to also tell them you do not have a new job lined up. This would trigger them to make suggestions or connect you with other companies. This is actually a great time to network, using your existing team. Hope this help, and good luck! Jim

So far it sounds like you have been passively accepting of anything that happens to you. A lot of the solution could lie in going proactive

Talk to your manager, in private. Tell him that although you are okay with leaving the company they should be the one to initiate proceedings. If they want you to leave, they should set the date. If they don't? Tell your manager that you will assume nothing has changed and that you plan to be professional about it until you find another job.

Why? Because this turns it from your problem to theirs. They thought they could get rid of you with the minimum of fuss, prove them wrong (note that just doing your job until you get fired counts as 'fuss' in this case, I do not recommend going postal.)

Now if worst comes to worst and you do get fired before one of your options turns up, you can benefit from any unemployment benefits. Explaining a "gap in your employment" is not nearly as big a thing in the modern economy as it once was.

In many industries in the US, it is socially acceptable to state that you would rather not disclose your next employer. That information is yours to share or not share in any country and industry, but YMMV as social convention may differ depending on those 2 factors.
EDIT: As you clarified that the industry is software in the US, I can say definitively that "I don't want to disclose that." is an acceptable and professional response to "Where are you working next?". In fact, I had done that at my previous place of employment, despite having already accepted a job offer. I did not want either a) persons following me to my new job or b) persons I didn't see eye-to-eye with perhaps acting unprofessionally (not that I would be worth the effort to them, I'm just paranoid about that).

That being said, if you don't have a job lined up, why are you resigning? Resignation is voluntary, if you don't have another job lined up, make them fire you rather than resigning, as in many jurisdictions, unemployment can be tied to the form of separation. Additionally, a verbal "You need to quit" that isn't backed up with any sort of HR driven process holds no water. To fire you, they will want a paper trail of the process to cover their tails in case you sue for wrongful dismissal (again, assuming USA).

If it's just disagreements with your boss, and no true violations of company policy or local legal codes, the proper move on the part of the company may be to move you to a different business unit/department/management chain.

I was in your position in October of last year, however I fired myself out of my own initiative.

I did not hide anything from the people I worked with; I just told them the plain truth, that I was tired some things were not working smoothly for a couple of years, that I had a couple of lined up opportunities but that would firstly take some longish deserved holidays with my wife.

I clearly do not see the source of your apprehension in having a work gap. Nowadays it is pretty common. Many people choose to took a sabbatical break and go in holidays, to prepare for a certification, or simply to work independently.

PS. I also mentioned in subsequent interviews I went out because I was not happy for a good while and nothing was done about it and as of the gap, was recharging my batteries in a couple of months gap, and taking a more wide view of the world travelling. All interviewers took my explanations fairly well, including my current job, except for one place where I was criticized for having an independent mind/opinion different of my upper management and travelling while unemployed. I would not like to work with such small minded people anyway, they seemed more interested in a desperate, low-paid serf than a smart, productive employee.

PS2. While we took a couple of months off, I have a workmate in my current job who told me he took one year off with his wife for travelling.