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I've done the trip myself. I bet you could get by with a 60 or a 50L without getting too crazy on the lightweight spectrum. My newest pack is an REI 60L I believe and have taken several multi-night trips with it without really stressing on every ounce too much.
I'd say you can go a smaller than you would on a deep backcountry trip because if you go in season there are a lot of roadside shops you will encounter as you crisscross the road, so you won't need to stuff your food pack. Some close or reduce hours before Memorial and after Labor Day so I would check before relying on them.
If you don't mind a couple peices of advice, the Shenandoahs stays above 3,000 feet so a cool spring night at sea level is straight-up cold. I was still trying to make a 30 degree bag work in mid-October and regretted it. Also, don't be too ambitious. It is nice to take a long lunch at the shops in the Shenandoahs to have a burger and a cup of coffee. You can't do that if you plan 18 mile days. Also, it does get busy so I you get in late the site might be packed. Don't press it.
I really enjoyed it, think you will too.