Speaking from the Virginia perspective. We have a lot of excellent state parks that offer solid hiking. The website is here (https://www.dcr.virginia.gov/state-parks/).
A lot of the easier hikes are in the Central Virginia area. My home park is in Chesterfield (Pocahontas State Park). It offers over 100 miles of trails. However, there is no primitive or backcountry camping.
I love, love, love the parks on the west side of Virginia. Grayson Highlands State Park hooks into the AT and offers both primitive and backcountry camping on AT. Great views, challenging hikes and plenty of open space. However, it's bear country.
Also, Shenandoah National Park is another big one. Lots of primitive and backcountry camping. It also hooks into the AT. I've only done a couple of hikes there. However, they were gorgeous. Waterfalls, cliffs, bears, views, etc. Unfortunately, the Old Rag trail is closed indefinitely. It is one of the most iconic hikes in the park. The on-going mess is still keeping the park closed. I would check the website (https://www.nps.gov/shen/index.htm) before planning a trip/hike/camp there. Skyline Drive is a beautiful drive through the park in the fall.
Just my thoughts.