Dyersburg is too small to have BAD neighborhoods and there is only one Walmart. I'm sure you would be perfectly safe there.
2 Walmarts in Jackson and they are both in good areas. A few poorer neighborhoods in Jackson but nothing really bad.
The Sam's Club & one of the Walmarts in Jackson are on the north side of I-40, very respectable area, close to each other. Either one would be fine for a quick overnight.
Cracker Barrel in Jackson doesn't have a very big parking lot, Walmart or Sam's would be a better choice.
