Check out Devil's Den State Park, Hobbs State Park, Blowing Springs, or Coler campgrounds. I've only ever camped at Devil's Den but I've visited the other parks and they all seem pretty nice. The state parks are certainly more out in nature whereas Blowing Springs and Coler are sort of in town, though they are in Bentonville/Bella Vista, so not particularly close to Fayetteville.
Normally Lake Wedington is one of the closer campsites to Fayetteville, but I just looked it up and the campsites are temporarily closed. You could also try Withrow Springs State Park near Huntsville, which is about 40 min from Fayetteville.