Should you move out of a region, you will need to contact Social Security (the administrative arm for Medicare) to notify them of your new home address. Under Medicare you are then entitled to a special election period to change your Medicare insurance coverage(s). It begins the month of your new move and ends 2 months after your move.
Medicare Advantage plans and stand-alone drug plans are all regionalized so will differ by State. Medicare supplement/medigap plans are defined by Medicare so do not change by region — however, not all regions cover all medigap plans and the premiums by plan will differ by insurance company and by region. Make sure to do your homework before you move so that you have time to review your new options before selecting coverage that will best fit your specific health/drug needs in your new area and to make sure you don’t have a break in coverage.