We just returned from two weeks in your neck of the woods. Every camp ground we stayed had lake/river access and all the grounds were very quiet and well maintained/managed.
Warriors Path State Park is few miles up the road from you.
Panther Creek is even closer to Knoxville.
Lake Barkley is a bit further down the road but still easy.
Piney Grove was probably the best. Right on the water which was rising daily because of the rains in the Tennessee River area.
All my nights were in the state and federal public parks and didn't spend more than $30 a night. Corps Of Engineers (aka COE) parks are going to be your best option considering they manage most all the Federal camp grounds associated with TVA dams. Most with power and water but you'll find some that are more rustic in nature.