I hope to add to this list, but for now it is just my favorite campsite that is within a 90 minute drive for me.
A relatively short but steep hike leads to a great view of the Blue Ridge Mountains at Stone Mountain State Park, a couple of hours north of Charlotte. There are also two waterfalls worth seeing. I made a six and a half minute video of my hike there last month:
Campsites are usually available any weekday, and most any weekend out of season. Reservations can be made at ReserveAmerica.com. I have never seen a fiberglass trailer there but I plan to remedy that next spring.
Main web site for the park: