Depending on your preference of staying at one park, hitting a few parks and/or if your open to other types of parks, here are a few options in the SE:
Bandits Roost, Yadkin River Valley, western NC:
https://www.recreation.gov/camping/b...O&parkId=73074
Kings Mountain State Park, Kings Mountain, NC:
Kings Mountain Hiking - Kings Mountain Campground - Kings Mountain State Park
Edisto Beach State Park, Edisto Beach, SC
Camping at Edisto Beach State Park
Good luck and enjoy your trip!