Let me state first: overweight and completely unfit, just started my journey a month ago. This isn't about weightloss or appearances in anyway.

I have serious problems with my posture and balance as well as myriad other problems. I had to do a consult with a physical therapist, who wasn't able to do much because it was just a consult and I'm not paying out of pocket to go back, and he basically said a lot of my problems could be fixed if I were to strengthen my back/core and balance. What are some good exercises/options/advice?

Extremely poor and live in someone elses tiny house with 5 other people. I can't bring anything else into this house, even if they'd allow it anything more than a handweight would be a tripping/falling hazard.

TL;DR What are good equipmentless excercises for core/balance/back/shoulders?