I've been looking into dropping out of my tech job specifically to start a consulting practice helping individual and teams through such things, because there are some common threads that everyone seems to struggle with. You said some of them yourself - areas out of your control, responsibility without authority, self-identity wrapped up in work. I hear those three all the time.
So you are not alone.
You are on the right track with how to cope - identify the problems and take some vacation time. The next step would be to discuss the problems with your boss - tell them what you need, and what you want. Tell them you aren't OK where things stand right now. Draw some boundaries about what is or is not OK in your work environment.
Also take some time to think about what areas of your work are truly your responsibility. Do people count on you because it is your job description to be the linchpin holding your team together? Or do they count on you because they know you can be counted on, without that actually being your role? I typically hear from people in the latter case - and when you bring it up with your boss, tell them that something needs to change - if you are the key person, your job title and compensation need to match that responsibility. You also need the autonomy to control everything you will be held accountable for. If they do not grant you those things, then you should step back and tell people to count on someone else, because they aren't giving you the tools you need to fulfill that role. Push the pressure you feel uphill to your boss, who is truly accountable for your team.
Those conversations can be tough. But in a situation like your where you don't hate the job, you just hate how it makes you feel, your employer wants the same thing you do (usually) - to make it a place where you want to work. Help them get you to that place by talking to them. Odds are, you'll work something out. If not, it is time to look for new work, but cross that bridge only if and when you come to it... for starters, just get the conversations going. After all, you have nothing to lose - if you don't fix this, you'll quit at some point anyway when the mental strain grows too much. So give them a chance to help you fix your job.
So you are not alone.
You are on the right track with how to cope - identify the problems and take some vacation time. The next step would be to discuss the problems with your boss - tell them what you need, and what you want. Tell them you aren't OK where things stand right now. Draw some boundaries about what is or is not OK in your work environment.
Also take some time to think about what areas of your work are truly your responsibility. Do people count on you because it is your job description to be the linchpin holding your team together? Or do they count on you because they know you can be counted on, without that actually being your role? I typically hear from people in the latter case - and when you bring it up with your boss, tell them that something needs to change - if you are the key person, your job title and compensation need to match that responsibility. You also need the autonomy to control everything you will be held accountable for. If they do not grant you those things, then you should step back and tell people to count on someone else, because they aren't giving you the tools you need to fulfill that role. Push the pressure you feel uphill to your boss, who is truly accountable for your team.
Those conversations can be tough. But in a situation like your where you don't hate the job, you just hate how it makes you feel, your employer wants the same thing you do (usually) - to make it a place where you want to work. Help them get you to that place by talking to them. Odds are, you'll work something out. If not, it is time to look for new work, but cross that bridge only if and when you come to it... for starters, just get the conversations going. After all, you have nothing to lose - if you don't fix this, you'll quit at some point anyway when the mental strain grows too much. So give them a chance to help you fix your job.