Cycling the Blue Ridge Parkway in NC/VA and Skyline Drive in VA would be a great alternative. Lots of cyclists do this. The parkways have relatively light traffic and are incredibly scenic, with many places to camp along the way. There's even a book about, the title is something surprising like Cycling the Blue Ridge Parkway. Google it.

Some other options are the C&O Canal trail, which runs from Washington DC as far as Pittsburgh, PA, I believe. That is unpaved if you prefer off-road. Also check out the Natchez Trace parkway, which runs through Mississippi, Alabama and Tennessee.