If you want to camp as close as Charleston, Oak Plantation CG is nice, and convenient to James Island as well as downtown. Haven't seen the KOA, which is on the other side of town (north, also near the shore) but have heard good things about it.
We stayed at the KOA south of Savannah for several days. Not super, but clean & friendly, and there are tons of things to do from there, including the downtown (lots of history - plenty of old homes to tour), historic Fort Pulaski, more beaches, etc. We even took a day trip from there to St Mary, where there's a little but wonderful submarine service museum in town.