The TVA Headwaters campground has W/E and a dump station. It is nice with decent showers. There is the Tailwaters campground too right below the dam on the river. Not as nice as the Headwaters but will serve your needs.
How far do you consider too far? From both CG's you are roughly within 10 miles of Pigeon Forge. Traffic may be heavy because its peak leaf peeping time here. You will be removed from the noise at these CG's. Anchor Down was mentioned and it is by far the nicest one in thearea but that comes with a price. Townsend is miles away from Pigeon Forge and G'burg, but is a wonderful area too.
While here go to the Island complex. To find it just look for the big Ferris Wheel. Its a nice place to walk around and eat at Paula Deens or Margaritaville and a couple other good restaurants.