Crowders / Kings Mountain is really close near Gastonia on the NC/SC border. The 20min hike is fairly easy and sites have designated camping pads, fire rings, grill, and lantern post. I think it is $5-$7 a night?
The sites are close enough to one another that you can see the neighbors' sites on either side but they also have free split firewood which is always a plus, a water pump, and an outhouse.
The hike to king's pinnacle is nice and on a clear day you can see the Charlotte skyline.