Will second Bama Ed on James Island County Park. We have stayed there, and it is very nice. But, a big problem!! It's so popular that reservations have to be made far in advance and very hard to get into during prime seasons. We have also stayed at Oak Plantation CC which is nice.
The DW and I have stayed at the Mount Pleasant KOA 3 times and have had good experiences each time. It is a short, simple and fairly direct drive into downtown Charleston. Bathhouses were clean with separate showering stalls along with fairly flat campsites. Lots of good places to eat nearby. Poe's Tavern on Sullivan's Island along with Red's Ice House on Shem Creek and Page's Okra Grill to name a few.
We love James Island County Park. Had a reservation for July that they cancelled. Received an e-mail stating they were canceling all reservations for 2020, would not be taking any reservations for the remainder of 2020 and suggested making them for 2021.