Just for the heck of it I clicked on the "US 340 N" option on Google Maps bike directions. I followed it's mapped route north from Waynesboro to find that it indeed follows an intricate network of country roads avoiding US 340 entirely all the way to Luray. It's a typical Google bike route that in reality would be hard to follow with all it's twists and turns, but it sure looks like it would be fun to try if you have the time. I checked a few random locations with "street view" and it looks nice.

I see only one short jog on US 33. All the rest of the way to Luray it is county and state roads. From there to Front Royal it follows 340. You could take the "Cat Back" mountain option I mentioned in my first post to connect to Fort Valley back to Front Royal. Warning: I called Cat back mountain a "stiff" climb. It's been over 40 years since I've ridden it but I remember it more as a "brutal" climb with switchbacks!

Google Route is 80 miles, Fort Valley would add some to that.