If you look closely, you'll find that about a third of all license plates in the North Georgia mountains are from Florida. Yes, we've finally figured out where Floridians go on vacation.
If you change to Georgia drivers' license, etc., you'll be liable for Georgia State Income Taxes and ad valorem taxes on your RV.
Just do like thousands of full timers do, and keep your Florida drivers licenses, license plates. And don't think about moving your personal business outside of Florida.
I keep my RV in Helen, GA, but my residence is in Alabama. Alabama's so much kinder to its citizens taxwise.