Between Ft Fisher, NC and Charleston, SC is Huntington Beach St Park. You can check availability online. Very nice park with both full hookup and water/elec only. Non full hookup sites tend to be larger. It is on the ocean with a short walk to beach thru dunes. You can't see the water from campsites. Also great place to bike ride and it's pet friendly. Leashed dogs allowed on beach.
Have not stayed there, but have heard the Mt Pleasant KOA is very nice. The location would seem to be much better for you than James Island which is a high traffic area on far side of town. The Mt P KOA is near updated highways/connector to Isle of Palms which has great beach and access to downtown Charleston area.