Previous posters are correct. If you are just headed south down the coast you will be a very long way from the Gulf Stream until you reach Hatteras or far south Florida
At Hatteras I think it's easier to go inside by the ICW
. It's shorter, nice scenery, and you don't have to sweat the weather
As far as when it's time to head
south to beat the cold, probably Sept to early Oct, although you can get an early cold snap even then. It is still hurricane season but you won't be going offshore
if you are headed down the coast so will be easy to pick weather
windows of a day to a few days.