Huntington Beach State Park, just south of Myrtle Beach, SC. It's much more pleasant than any of the public or private parks in Myrtle Beach, and is only about 10 minutes away from all the activities in Myrtle Beach. They take reservations and have some full hookup sites. RV access is very easy.
Julian Price Park, Blowing Rock, NC. It's a national park, on the Blue Ridge Parkway. It's a very quiet and scenic getaway, probably best appreciated by adults. There are no hookups, but at about 4000 feet elevation you'll never need air conditioning. RV access is not as bad as you would think by looking at the maps. If you come from the east (I-95, to I-85, to NC highway 421, to the Blue Ridge Parkway), access to the park is easy. We have done it with a 32 foot travel trailer with no problems. Just remember, the gear you go up with is the gear you need to go down with. The Parkway south of the campground is, in my opinion, too steep and curvy for large motorhomes or trailers.
Big Meadows Campground, Shenandoah National Park, Virginia. Within 2.5 hours of Washington, DC, a beautiful campground on the Skyline Drive. It's busy on summer weekends and in the fall. No hookups, but you don't need them. RV access requires careful driving.