We always camp inside the Great Smoky Mountains National Park, our strong preference is Elkmont Campground. You are dry camping here though, so no hookups at all. You are a very short hop into both Pigeon Forge and Gatlinburg, but staying out of the craziness.
If I HAD to be out of the park, I would go to the Townsend KOA, about a mile from the entrance to the park between Cades Cove and the Sugarlands Visitor Center. If you stay here you are away from Gatlingburg and Pigeon Forge, and all the touristy stuff there. Full hookups and right on the river.
Our family much prefers the activities inside the park to anything in Pigeon Forge, though. We spend about 15 total camping days there every year.