I was coming back from a medical leave, before I left for medical leave I was asked if I could give one day availability and I agreed. After trying to return, I had gotten a new job that was full time, my new supervisor was trying to be flexible, he told me to speak to my other manager to see what days she needed me. She did not get back to me for two weeks and my new manager could not wait for her, so he had no choice but to give me my set schedule. When she finally did get back to me she wanted me available for times that I could not do due to my other job. She also told me that I need to give four days for availability, when in the beginning she asked for one. If she had gotten back to me in a timely manor I would have been able to accommodate her needs. Now she wants me to resign.
You should probably call an attorney, an employment attorney should be willing to give you a free consult. The facts could use some more development, which would be rough in this format. One question I would have is what is the nature of the condition, was it caused or exacerbated by work. The other possible issue could be the handling of the FMLA leave and any ADA issues.
Never resign. You would be ineligible for unemployment and your employer probably knows it.
The circumstances where you have another job are always difficult because they have no duty to try to accommodate another employer's hours. You really need to consult an employment law attorney. Your situation sounds complex and it seems like you are entitled to FMLA leave. You have laws that seem to protect you in a certain fashion but it is a mistake to go it alone and not consult an attorney with all the facts because particular facts make a big difference in the advice you will receive.
Employment / Labor Attorney
This is a common tactic with Employers. They will attempt to get you to resign so that the burden would be on you to establish that you left the job for a necessitous and compelling reason in order to obtain unemployment compensation. You should avoid signing anything that says you are resigning. You should also make every attempt to work with the employer to establish a mutually satisfactory schedule. If you are unable to reach an accomodation be prepared to defend why you could not work the employer's schedule showing that you exhausted all available options.
Employment / Labor Attorney
While you can still get unemployment benefits if you quit your job, it just makes it that much harder for you. When you quit a job, you have the burden of proving that you had "good cause" for quitting. Your employer is aware of this fact. That is why they are trying to get you to quit rather than firing you. I would advise you not to resign.
Instead, I would work with your employer on trying to come to an agreement. Try to be as accommodating as possible.