Oh, I know this feeling well! I had the same thing in my last job. I actually got made redundant earlier this year, so I'm not working at the moment, but it's taken me this long - and not being there - to realise just how stressful I'd found my working environment. And that stress was largely down to lack of privacy/personal space.
It was a large, very busy/buzzy open place office and all the meeting rooms had glass walls - you couldn't get a moment of privacy anywhere! Nightmare! And I sat In an area with about 20 other people - I liked them all and got on with them and none of them were particularly loud or disruptive....but I got to the point of just finding their proximity unbearable. It was nothing they were doing wrong - I just couldn't stand to be that close to people and increasingly started to feel that I needed to escape.
Like you (and many others here, I suspect) I'm a person who needs alone time to recalibrate and I was finding that in my last few months there I was hardly spending any time at my desk - I was trying to find corners of different floors of the building to hot desk at and I spent a lot of time in the loo, talking to myself in the mirror to calm myself down or just sitting on the loo zoning out to find some peace. In retrospect, these wouldn't have been especially good strategies long term as they were really just about avoidance and hiding. But they did allow me a much needed breather at the time. I should say that I'd worked there for some years and, while I always found the environment a bit tricky, it wasn't as bad as those last few months - I think it got so bad because my PTSD symptoms got worse, so I was incredibly anxious/hyper vigilant/always feeling on the cusp of panic, so the office environment just compounded those things.
If I get another job in an office like that, I'll definitely be more mindful of putting better coping strategies in place - in fact, I was working on some of these ideas with my therapist before I got made redundant. We were discussing things like: having a 'safe' place to go if I just needed to get away and have some alone time (in the office or outside; building regular short breaks into the day so that I could go and do some breathing exercises/meditation or even just go for a walk to a different floor to get a cup of tea from a different kitchen; where possible, going for a short walk on my ownoutside at lunchtime.
I don't know what sort of work you do or what your working environment is like, but I think the only real solution is that you need to be quite disciplined and commit to making the time to do these sorts of things - start small and look for ways you can fit in a couple of ten minute breaks in the day for instance and then try to build it up. I do understand it can be difficult to find the time for this sort of stuff - especially as you're fairly early into a new job - but, in retrospect, I really believe that if I'd have put these sorts of things in place, it would have really helped me.
Good luck!