Your heading seems to contradict what your post says. Supported tours tend to be groups of cyclists riding the same routes, often hundreds of them.

However, based on your actual post, you might want to check out the C&O Canal - GAP trails that run from Washington DC to Pittsburgh. It is close to civilization, hotels, campgrounds and food, but you are riding on a trail and can camp out if you want. That would only take 5-8 days, depending on how far you ride each day. Some longer options might include the Blue Ridge Parkway and Skyline Drive in NC and VA, but those can have lots of cars on weekends and holidays. Natchez Trace in Mississippi and TN is similar. Or you could go to Wisconsin for a few weeks and patch together a tour riding rails-to-trails and rural roads.