Several suggestions for your trip.
We are heading to Chat. in April and plan to stay at Chester Frost Park which is operated by the county.
RV Park Reviews :: Hixson, Tennessee ( TN ) - RV Parks and Campground Reviews
This park does not take reservations. There is a big festival in Chat. called the River Walk or River Front Festival so you should check the dates you will be there. It is a lot of fun and good music.
In Nashville, I would recommend a COE campground - Seven Points. It is very convenient to the city and all the attractions. Since it is almost all full, you should make reservations asap is you want to stay there. Nashville is a great city - make sure you see the Country Music Hall of Fame. I enjoyed it even through I am not a big fan of country music.
Have a safe trip and enjoy TN, it is a great state to camp in.