Suggest you check out the KOA on the north side of the Cooper River. We definitely are NOT KOA people but this is an exception. Lots of good eating places on and around Shem Creek. James Island is several miles from downtown Charleston and the traffic is not kind getting over there. Last time we stayed there, there were overhanging limbs on both access streets getting there. I shudder to think if it were raining and the limbs hanging any lower.
Beaufort and Hunting Island SP is interesting (IF) you can get a reservation. Beach is great for kids. Light house is accessible and there is a huge fishing pier.