There are some very remote places that would be fun to tour at a slow rate. However, I don't believe either the Appalachian trail, nor the Pacific Crest Trail allow bicycles. Possibly a dog trailer could be used.
There are some long off-street bike path projects. Some might be remote enough where you would feel comfortable camping.
Perhaps look at some of the gravel road/trail adventures.
The West Coast has more public land than the East Coast, and thus more places for camping by the road. You would still have to deal with dog food.