Just be aware that getting to Western North Carolina from the DC area is about an 8 hour drive. As somebody who used to live in DC and had to make that drive every month (and it is a slog), I would recommend staying within a 3 hour drive of DC. There is so, so much in that radius to see based on your specifications: Skyline Drive and the Virginia Creeper Trail in VA, the C&O Canal Trail heading north out of DC (be sure to make it to Great Falls), Gettysburg PA, Harpers Ferry WV, Harrisonburg VA, Richmond VA, Shenandoah National Park…all with excellent riding.
But if you’re set on coming down to NC, I’ve got some recommendations!