Our circumstances sound similar. Only difference is I don't even think about relationships at all any more because I'm certain it's not worth it for me, and I wouldn't feel comfortable inflicting my baggage on another person anyway.

Unfortunately I don't even have a handle on my own mental health issues so the only real advice I can offer is on losing weight, because that's one thing that, if not easy, is simple to do. What helped me was buying a set of adjustable dumbbells and a weight bench (for about $100 total after shopping around for the cheapest ones with decent reviews), and setting it up in my room, so that I didn't need to leave the house to work out. That helped because A) at one point in my life I'd self-isolated for so long that any human interaction would turn me into an anxious wreck, and B) it lowered the minimum effort required to work out from "I have to get up, take a shower because I haven't washed in days, clean myself up so that I don't look like a schizophrenic homeless man, then leave the house and walk to the gym (because I couldn't afford a car)" to "all I need to do is get out of bed." Diet is more complicated, but it's a good idea to limit your carb intake and get most of your calories through fat and protein, because they're more sating than carbs, so you'll feel less hungry. Look up a TDEE calculator and count your calories, and you'll be mathematically guaranteed to lose weight.

I think the ideal thing to do would be to go to the gym, because I know from experience that going without human contact for too long can fuck with your head, but I also know how daunting a task that can be when you can barely muster the energy to do anything more than survive until tomorrow. Plus it'll save you money in the long-term if you can afford the initial investment, and you won't have to worry about membership fees, which is huge when you lack regular income.