You have 4 state parks that all have campgrounds, Myrtle Beach, Huntington Beach, Edisto, Hunting Island. Check the Atalaya Castle out at Huntington Beach. Across U.S. 17 is Brookgreen Gardens. You will be near Georgetown which is cool little town and about 90 minutes from Charleston.
In western South Carolina you have a series of State Parks in the mountains and one USFS campground, Cherry Hill, that is nice.
In Western NC, Asheville is a great place with a revitalized downtown. The Boone/ Blowing Rock, NC is nice too. ( higher elevation so it is cooler there)
Blue Ridge Parkway runs near all of these places. Mt. Pisgah Campground or Linville Falls Campground. Brevard, NC is a nice area and the DuPont Forest is where the Hunger Games movie is filmed.
Great Smoky Mountains National Park is in both NC & TN. we like Balsam Mountain Campground. It is small but closes early due to it's high elevation. Smokemont is nice and near Cherokee, NC. Cataloochee is beautiful area, the campground is nice but you do go down a windy road and 3 miles is gravel.On the TN side Cade's Cove is beautiful but can get crazy with traffic.
If you spend any time in TN, check out the Big South Fork National Recreation Area. It is on the Cumberland Plateau and won't be as crowded as Great Smokies. TN has a nice State Park system. Cumberland Mountain and Falls Creek are the two we have been to.