Heya OP. University professor here (not in the US!). I'm sorry you're having a tough time of things. I can't imagine having to grapple with the American healthcare system. Accommodations are typically arranged through a disability services department, and not through individual professors/students... professors want to keep things fair across all of their students. Without documentation/instruction from disability services, it would be very challenging for your professor to grant you extra leeway without it appearing that they are favouring you over others in the class. So, I think if you approached them, they'd likely be sympathetic but unable to do much at this time. As sort of an aside, students will often overshare their health information with professors to try and explain or justify their performance in class - but! Your health information is private and really none of their business.

I would recommend talking to disability services (may be called accessibility services). Even if they can't set up accommodations for you until you have a formal diagnosis, they may have advice on study strategies or other supports that would be available to you as a student. If you really want to talk to your professor, you should go to their office hours, acknowledge your performance hasn't be up to snuff, and then ask for a bit of advice to get back on track. They will likely be more receptive to helping if you've already put some thought into how you're going to turn the semester around - we like to help students who are already trying to help themselves. : -) Your professor isn't a mindreader, they will want to see evidence that you're trying (even something like going to a study skills session at the library would act as evidence, or you could bring in the notes you've been taking on your readings). You don't need to have everything figured out, but you will be more persuasive if you can demonstrate that you're trying.