What I am about to describe to you are GA/AL ("1" & "2") and, also, Virginia - North Carolina ("3") venues. I mention this so that if this geography is NOT of interest to you, you will not need to spend any of your time reading further.
1. In GA, west of Atlanta to the GA/AL border is a completely traffic free paved route entitled The Silver Comet Trail (SCT). Your campsite on the link/map diagram that follows is just east of the town by the name of Rockmart... the campsite is wide-open and friendly to cyclists and is called The Rock.
The Rock: www.therockrvpark.com
2. At the state line the SCT connects with another paved trail that runs from the GA/AL line to just north of Anniston, Al (Anniston is an exit on I-20 and you will find that it is located east of Birmingham, AL.). It is entitled the Chief Ladiga Trail. I can not speak to the campsites along this trail. www.chiefladigatrail.com
3. Forgive me if I missed any above reference to The Skyline Drive / Blue Ridge Parkway, so I will go ahead and suggest this, as well, for your consideration (NOT a winter ride - but this is now March :- ) ..... with the exception of Roanoke, Asheville and the southern terminus vicinity of Cherokee, I found it to be only about 4 vehicles an hour. This route will take some serious research for campsites, as well as, where to get on an off....not to mention re-supplying. This route is classified by most as arduous....at the same time it is as beautiful as any slice of Heaven. There are MANY sites for your research... here are the ones from the Natl Park Svc: