OK, to answer the question that you posed.....by the middle of August, a lot of the schools have begun, so the crowds that you would get during July will no longer exist. That said, many people will still be out on the weekends. As long as you don't plan to go to one of the MAJOR tourist areas (Pigeon Forge/Gatlinburg), you will probably be fine. I would just make sure to pull in during mid-week and check on weekend availability. Labor Day might pose a problem, so I would probably make a reservation soon for that.
For the days in each area, I would just play that by ear. You might find some things that interest you and you wouldn't want to rush that since you have no time frame.....
I really don't know of any campgrounds in/around those areas, so someone else will have to help you with that.
Just for your research, you might want to consider the Tri-City area - Bristol, Kingsport, Johnson City. It is also very "interstate" friendly, but has a lower cost of living than some of the more metropolitan areas. Plenty of lake/water (TVA) areas, more outdoor activities than the metro areas, but still have universities, symphonies, great hospitals, shopping and other amenities.