I rode from Front Royal to the end of Skyline Drive plus the Blue Ridge Parkway to Buena Vista last July. Then I cut further west and came back north in or near George Washington National Forest. It was a great ride, nice scenery, little traffic. I found a map series that covered the entire trip in 4 maps and planned much of the route sans Skyline and the Parkway on the fly. I got the maps at REI.

Long story short - go futher west than 340.

You can park your car at the Front Royal Fire Department and they'll keep an eye on it for a small donation.