I would say without a doubt most if not all the various Southern accents were around during the CW.
One note I'd like to caution people on, is the Scots-Irish connection. Its very small, I would say outside of Appalachia its very rare and that most Southerners are English decent, and that Deep South accents probably have more similarity with English ones.
Some may balk at such a notion, (I know a lot of people that hate me for saying it), but when you think about it makes sense. People tend to immigrate to regions similar to where they came from, you have that in Appalachia for Scots-Irish, and on top of that Scots-Irish were poorer peoples, and tickets to the South were more expensive, the voyage longer distance than say immigration to the North from the British Isles. Then there's the American Revolution to consider, during that conflict there were a lot of people for the King in the South, and as a rule they all tended to come from people of Scottish, and Scots-Irish settled areas that were in direct conflict with Patriot areas that were mostly English decent. A lot of those "Tories" were forced out after that conflict.
Then there's practical application. I speak with a pretty strong Southern, Texas drawl, and I will say with full confidence it has more in common with English accents than Scottish, because if I sit and watch British shows, (I've always like the Napoleonic shows), for an extended period, I've found I can replicate various accents to near perfection while hearing it. I've shocked some folks that way for fun I might add. I can't do that with Scottish, or Irish accents, plus I've seen folks from England who could replicate Southerner accents to near perfection, but I've personally not seen it done out of Scottish or Irish peoples. So while a bit circumstantial, I think that ought to be brought up.
Of course then there's surnames, which tend to either be complete English origin in the majority, surnames that'd become interchangeable with England and Scotland, and then Scottish surnames in dead last outside Appalachia. I'm by no means an expert, these are just things I've noticed. Also Southern newspapers at the time of the CW, and before like the Charleston Mercury, defined Southerners as Anglo-Saxon decent, little food for thought.