This is my suggestion and may not be the best solution.
Travel east out of Damascus, VA on US 58 and follow it to Mouth of Wilson, Va. This is about a twenty mile ride perhaps and I believe most of this is still part of the Trans-am route. When you reach Mouth of Wilson you are essentially on the NC border. If you turn down VA 93 south you will cross into NC within a mile. This is the western end of NC Bike Route 4 which runs along the northern border of NC. NC Bike Route 4 essentially connects to the Trans-Am at this location because the distance between the two is just a few miles difference.
Follow NC Bike Route 4 until you reach NC Bike Route 1 near Kerr Lake and Henderson, NC. Turn south on NC Route 1 and follow it to Apex, NC. In Apex you will intersect NC Bike Route 5 which goes to Wilmington, NC and follows the Cape Fear River. In Wilmington you can pick up the ACA Atlantic Coast Route which can be followed to the Charleston area.
Maps are available from the NC DOT for the bicycle routes and the routes generally have direction signs at the intersections. However, signs can sometimes be missing. The route is not the best as it passes through the Raleigh area on Bike Route 1 as the route was planned many years ago and Raleigh has grown significantly.
The web page for the NC bike routes is: http://www.ncdot.gov/travel/mappubs/bikemaps/