Welcome to the RoadTrip America Forum!
Your time is sound - it's only a couple days trip between NYC and Miami, so three weeks total gives you plenty of time to get around.
The mountains of Virginia and Kentucky, some of my favorite places, are in your path. There certainly wouldn't be any city life partying in those areas, but if you're looking for a couple of days to spend out in the country and to relax a bit, I couldn't think of a better place to be.