Been there done that!
Have camped at Hilton Head, Charleston and Myrtle Beach.
Hilton Head, not a lot of options. The one you are going to is great, it is a marina and campground just over the bridge to the island. We enjoyed our stay there, but not real big HHI fans. Another barrier island to consider, although somewhat further south is Jekyll Island - lovely "in the woods" campground there at north end of island, and island itself is far less commercial than HHI with beautiful bike trails all around island.
As far a Charleston is concerned, in our opinion the only place to stay is James Island County Park. The campground and park are fantastic. But can be difficult to get in during busy season. They have a very liberal cancellation policy though, which allows cancellation up to day before, so we always get on waiting list. They do follow up and I have rarely been dissapointed. If you have kids they may enjoy the climbing wall and water park, both on the property. Charleston itself, so much to see I do I wouldn't know where to begin. Take some sort of tour - walking or buggy ride. And getting around in town is far easier on bikes than any other form of transportation. We generally park down on the battery and bike around town. Must sees, in my opinion - the straw market, the battery, the park on nort side of town, rainbow row... great restaurants and "watering holes" - do not miss the stuffed hush puppies at Fleet Landing. If you want a great dining experience ask for directions to The Wreck, great local experience, north of town a few miles but worth the trip. My son lives and works in downtown Charleston so much time is spent there!
As far as Myrtle Beach. We prefer Myrtle Beach Travel Park, it is on north end of town and right on ocean. Area is a little less congested than the mega campgrounds on south end of town, i.e. Ocean Lakes, Pirate Land, etc... they are nice to but in very busy congested areas. Sometimes line 2-3 miles long to get into the parks on hwy approaches! Myrtle Beach State Park and or Huntington Beach State park are great alternatives, do not have the amenities of the big parks, and are much more "woodsy" with little hike to beach. Size restrictions apply to, can't tell if you are in Motorhome or TT. Love the state parks for the feel of "camping" like I did as a kid!
PS - I don't know anything about MB in summer - I NEVER go between Memorial day and Labor day!!!! Did that once about 20 years ago, It is kind of crazy. Just like I never go to Disney during school breaks or holidays - to NUTS for us.!