This was such an enjoyable place to spend a week. We stayed in the Live Oak loop, which I recommend. The campground sits right in the center of an enchanted palm-frond forest with live oaks draped in Spanish moss. The sites are really private. We felt very safe here and let our kids bike around on their own, which they loved. Lots of hiking trails right off the campground. It's about a mile to the beach (you can bike there, but I'd hesitate going with kids, as the highway gets pretty busy and the bike lane is nothing more than a narrow shoulder).
There is a campground right next to the beach, which looked lovely but stunk to high heaven of decaying marshland. Water also smells, but is drinkable (though I would recommend bringing bottled water for drinking, especially if you have a sensitive digestive system). WiFi is non-existent, but phone signal is good, so if you have a Jetpack, you'll be just fine for internet. The beach itself is really lovely with nice, fine sand. There's a little corner store with grocery basics and a liquor store attached. Also a gas station not far from the campsite.
It is about an hour drive to Charleston and centrally located to a lot of nearby travel hotspots and other islands.