If you are trying to find your own quiet spot Chuck Swan has you covered. Lots to explore here, beaches, hi clearance roads(Jeep roads), caves, old cemeteries, rocks to jump off into the lake,and an old fire tower you can climb for what I would expect is a great view. This place was great and definitely needs more than a day to explore properly.
I drove to the end to The Forks of the River picnic area. This spot was absolutely gorgeous and secluded aside from the occasional boat passing by a ways out. The picnic area was very clean and had two picnic benches far enough apart to keep 2 different groups happy. There is a nice rocky/sandy beach to play in and it stays shallow for a ways out so it would probably be nice for the kids. I was pleasantly surprised how clean the area was, no trash on the ground and 2 cans that didn’t have much in them.
The main forest road is a gravel road that anything minus a sports car can handle. All other roads are not as well managed, and some are just double track dirt roads that you’ll need 4WD and possibly a winch if it’s muddy or rainy. I visited for an afternoon while spending the week a Norris Lake. Speaking to the locals the the only people that use it are hunters during the hunting season. Definitely check with the TWA for hunting info before you go as it’s an active area during the season. This whole area was a booming river town before Norris dam was built and there are remains scattered throughout the peninsula.