I have camped at Pisgah a couple of times. I love it, but it is the forest primeval. Gorgeous and wild. There are campgrounds with amenities, and there are rougher areas. Pisgah National Forest is one of my favorite camp areas. It's so beautiful and peaceful.
The thing with Pisgah is that the sites are pretty small but you're right next to the best hiking on that stretch of the Blue Ridge Parkway. The BRP is under construction around that area until the spring, so be sure to keep an eye on that. I plan to go back soon. I hiked out of Carvers Gap. Looks like good camping up there as well.
For a short hike in with a hike loop I'd recommend John Rock. Panthertown Valley is an incredible place. Shining Rock Wilderness is awesome but was closed to camping last year due to bears. I'd also look into Jones Gap State Park - it's just over the state line in SC.
We get a lot of rain during the spring. So if you choose the mountains, check the weather first. There is also primitive camping at Graveyard Fields, but it's been closed this year due to the bear activity (lots of blueberry bushes there). There is a restroom at the parking lot at Graveyard Fields, but no showers. There are also a lot of hikers in the area so you may have to hike in a ways to get some privacy.