You don't ask a question with regard to this situation, but I assume you want to know whether a judge will grant you a long continuance so that you can leave the state and help your family tend to your mother.
This is something you should take up with your lawyer so that he can make a request of the court for you. In my experience, that will have a better chance of working than a letter that you write to the judge. If you insist on writing a letter, however, make sure your lawyer sees it before you show it to the judge because if you make any admissions in that letter, they can be used against you and can harm your case.
Make sure that you have medical reports or some kind of hard evidence to show your mother needs critical care, even if it's just making it possible for your lawyer or for the judge to discuss your mother's surgery and prognosis with her doctor. With that kind of evidence, it is sometimes possible for your lawyer to get your appearance in court excused for a date or two in order to let you leave the state and tend to a family emergency. You will not be able to remain in your mother's state over the long haul, however, unless you start appearing on your court dates again. In my experience, judges won't excuse your appearance indefinitely, as you will always remain under court mandate until your case is over.
As for whether the judge will relieve your public defender and appoint a new one for you, usually they will allow for one change of assigned counsel if there isn't a good fit between the attorney and his client. But it depends upon how close to the end of the case it is. If the judge is going to look at this just as a delaying tactic so that you can leave the state, not only won't he change counsel, but he won't excuse your appearance in court either. That's why the best way to proceed with this is to let your lawyer make the argument and to give your lawyer the evidence that he needs to get your request granted.