We recently visited the area in October. East of Abbington is a town called Damascus and east on 58 we stayed at Beartree Campground. Quiet, well separated sites, showers, dump station, however, no hookups. The Viginia Creeper Trail runs form whitetop mtn back through Damascus and into Abbington...a real nice bike ride.
I was told by a local that there were two or three campgrounds on or near Holston lake just south of Abbington.
Anyway, it's a nice area. Hope you have a nice trip.
All the best.