There really aren't any State Parks or COE parks close around the "main" entrance to the Smokies which is the Gatlinburg/Pigeon Forge area. However these are also the very "touristy" areas. To avoid the traffic and craziness, go to Townsend which is about 20 miles away. There is also an entrance to the Smokies there and there are some nice private campgrounds.
Here are a few. Townsend is still close to all the trailheads that are near Gatlinburg.
I do not recommend Cades Cove campground. It's fairly crowded, and many of their sites are short or hard to get into.
Amateur Radio WA4GIY
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