1-2 hours south of Memphis are some very nice Corp of Engineer Parks. Some at
Aratutla Lake, Enid Lake, Sardis Lake and Grenada Lake. We stayed at Enid Lake at Wallace Creek campground and checked out Persimmon Hill. Both were very nice.
Both had paved/cement pads, water and electric, bath and showerhouse and a dump station, nice size sites, grass, trees. Not primitive, but woodsy. There are several campgrounds on each lake and some are primitive if that is what you prefer. They are all off I-55 from 1/2 hour to 2 hours south of Memphis.
Enid Lake is 10 miles long, 2 miles wide and there is a boat ramp. Supposed to be excellent fishing.