You did not say what kind of nurse you are. Are you an LPN? An RN? or some other kind of nurse classification that I may not know? It might be helpful to know about your training for any of us to give you a really good answer. I will speak from my experience with a nurse, so take it all with a grain of salt. :-)
Mrs Media was a nurse for 42 years, so although I was not a nurse, I've lived with one a long, long time.
Her nurses' training program was in a "diploma school," attached to and part of a hospital. She went year round for three years, and her training included shifts in the various departments of the hospital, so she was in the ER, the endoscopy department, the psych department, the morgue, the med/surg floors, outpatient, cardiac care, etc., etc. It was a very thorough training with lots of hands-on experience. She graduated as an R.N. (registered nurse).
Since that time, she has worked with lots of LPNs (licensed practical nurses) and new nurses who did not have the same intensive training she went through. Nearly all the RNs today graduate from a college study program; many of them get degrees, etc. When they came to the hospital where Mrs Media worked, she would often have to nearly "re-train" them. While they had WATCHED how an IV was administered, and while they had PRACTICED putting an IV needle into an orange (or something like that), they had never actually administered a "real" IV to a live patient. Mrs Media had to show them and watch over them, etc. Other new nurses weren't so lucky; they were "thrown to the wolves" and had to learn on the fly.
From what you wrote, the hospitals where you live take on these newly graduated nurses and enroll them in some kind of in-house training program, or, as you called it, a residency program.
Since you were enrolled in such programs with some hospitals in your area but did not complete their residency training--whether you had good reasons or not--they don't want to re-admit you to their programs.
My suggestion is that you sit down one-on-one with an HR person, explain what happened in your previous attempts, and -- rather than ask for them to give you another chance even though you know the answer will probably be "no" -- ask her/him what the best avenue is for you to move forward in your career as a nurse. That way, instead of asking for a job, you are asking for guidance. You are asking someone to give you the right advice instead of a job. People will be much more willing to make suggestions for you that way because they aren't being asked to hire you. You may actually find one who will not only point you in the right direction but who will also make a phone call for you to help you along the way.
There are also non-hospital nursing jobs at clinics, temp nursing facilities, health care facilities, and doctors' offices. You might try those avenues if all the hospital roads are closed.
Best wishes to you as you pursue a deserving career.