A. If you have Original Medicare, you can get a new Part D plan without penalty if you do it in a timely way. If you get your medication coverage from a Medicare Advantage (MA) plan, it's a bit more complicated.
You will likely need to switch to a new plan for all your coverage, as plans often do not cross state lines. The timetable is the same for original Medicare or an MA plan: your special enrollment period starts a month before your move and lasts until two months after the month you move.