Honestly, this relationship is so new, I wouldn't stay in it. I have MH issues, and so does my significant other, but both of us actively work to stay healthy. I couldn't be with someone who wasn't working hard to be stable and reliable (whether this means working hard in therapy, taking meds, actively working on self-care, whatever). Your "old boyfriend" is only a part of who he is; what you are seeing now is not an aberration, but another important part of who he is. You can't make him take care of himself; he will have to want to do that.

But everyone is different. What I need may not be what you need. But I think you should think carefully about the long term, and whether the caretaker tendencies you seem to be displaying here are ultimately harmful for you. Good luck.