We enjoyed our stay at Oak Plantation CG, south of Charleston. It was still easy to reach the Battery area and the Mt. Pleasant area too (we loved Patriots Point - but we also were Navy.) Great park with several Navy & Coast Guard ships you can tour. Anyway, Oak Plantation was easier to reach with a large TT than James Island, and had shade, grass, space between sites (at least in some of them, including the area where we camped). It's near the Angel Oak (also worth a visit).
Other good places to visit/things to do in Charleston: definitely do not miss a walk around the Battery area. Charleston is a gorgeous and gracious old city, and the Battery is the heart of the historic area. Middleton Place and Magnolia Gardens are good too. There are some county parks which are beaches.
Edisto Island SP is south of town a ways. Nice park, and you can camp quite close to the beach. Very atmospheric, with lots of Spanish moss, but I'm sure the bugs can be thick there. (We camped there many years ago in the winter - in tents. Remember, we were Navy once.
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2020 Toyota Tundra CrewMax 5.7L V8 w/ tow pkg, Equal-i-zer
2020 Lance 2375