Peak leaf color is the 3rd-4th week in October. Even if you can't get a site at Montgomery Bell State Park, spend some time there. There are lots of trails for hiking, historical interest, and beautiful scenery. (I live close by.) Cedars of Lebanon State Park is east of Nashville, and very nice.
Also go on ReserveAmerica or RVparkReviews and look for the COE campgrounds in the area. They are very nice.
You will NOT go wrong spending time in Tennessee's great state parks.
Take a day and head west up I-24 to Land Between the Lakes. It's a Forest Service facility (formerly TVA) and has a ton of cool stuff, including the elk and bison prairie, a restoration of a 1850's farm, and much beautiful scenery.
If you like history, the area around Nashville and Franklin is rich. President Andrew Jackson's home, The Hermitage, is in Nashville. There are several well-preserved ante-bellum homes in Franklin, TN and an entire battlefields (Stones River) in Murfreesboro.
The Natchez Trace Parkway begins just west of Nashville. Take it down through Tennessee for beautiful drive with waterfalls.
Single empty-nester in Middle TN, sometimes with a friend or grandchild on board