Skyline Drive and the Blue Ridge Parkway are great roads for sightseeing and even a slightly spirited romp. The speed limits are pretty low and there will occasionally be a Park Ranger to hand out performance awards so the best strategy for these roads is to go quick in the corners but don't hammer it in the straights - just tootle along no more than 10 over. This is a great ride and highly recommend it.
That said, the absolute best roads in Appalachia are found in West Virginia, Western Virginia, Western NC, and eastern Tennessee and they don't have catchy names involving snakes and dragons and such. We all have our favorites. There's way too many to list here and I'd recommend buying a copy of the Butler Maps for the region and going on a twisty safari and let it unfold serendipitously. There's all manner of roads, tiny towns, and interesting people if you leave yourself open to the meeting things as they come.
Or better yet, figure out how to get one of the really experienced riders on here that frequent the area to lead a ride. Unfortunately, I'll be headed to Durango about that time to ride the COBDR and the Continental Divide on a KTM 690 Enduro.
'12 K1600GT (gone but not forgotten)
'13 S1000RR (sold it to Allstate)
'14 KLR 650 New Edition (gone and thanks for the memories)
'16 R1200GS Triple Black (for all of it)
'16 KTM Super Duke 1290 R (for the inner hooligan in me)
'18 WR250R (for all of it...but slower)