Shenandoah is bear central, and I think you need a minimal permit. If you're terrified of bears pick a different section--most of VA has bear hunting and additional bear-dog training seasons. Find one of those areas and you'll probably never see a bear.
Maybe I'm partial to the area I live in, but maybe south/central VA might be a better section for you. Lots of neat stuff to see.
Hiking from Damascus north to pearisburg would be a good section. There are wonderful people that operate low-priced shuttle services out of Damascus that you can probably locate on whiteblaze.
Park at the rendezvous motel in Pearisburg, VA. Give them $10 or $20 for looking after your car, and get the shuttle to drive you south to Damascus.
Hike north to Pearisburg. Send a resupply to The Barn restaurant where the trail crosses 81 (Rural Retreat, VA)