Leaving from GA has few advantages. Due to the trend of the coastline you are about 100 miles further west leaving from GA than from S FL or SC. From GA you could
head east until north of the Abacos and then turn south. If you try a more direct
route you will be sailing against the
Gulf Stream a lot of the trip.
Charleston is much further east and could give you a good reach all the way but the
Gulf Stream could still be mostly against you. You could cut across at an angle to get past the stream but that adds distance to the trip. If you try to cut inside the stream you end up sailing down the coast and give up the easting you would get from SC.
I would pick Charleston as the better option.
Regarding time, if you are following the standard summer in
Newport,
winter in the islands or similar pattern then the time would be after
hurricane season but before
winter storm season. This, in my opinion is a gamble demonstrated by the
history of even well found
boats that are occasionally
lost heading south in Oct-Nov. The odds are very good but hurricanes are not unheard of in Oct or even Nov and I have seen winter storms in Sept.
If you are choosing strictly on the basis of the best/safest weather then very late spring before
hurricane season kicks off is my opinion (I assume you know what opinions are worth). However, to get a good
forecast for a reasonable five day window should be doable most times these days, even in hurricane season.