Hey Brigid, not a huge help on route as I have been traveling more northerly to Colorado and points north. Having a RTT you will be wanting a quiet place to camp. One I found that may be close to your route is just off I-22 in northern Mississippi in Holly Springs NF. Chewalla Lake Rec Area.
Explore Chewalla Lake Recreation Area in National Forests in Mississippi, Mississippi with Recreation.gov. Chewalla Lake Recreation Area is located in the Holly Springs Ranger District in northern Mississippi. Visitors enjoy swimming and relaxing on the sandy beach o
www.recreation.gov
I was on my way to Arkansas at the time and it was a very nice stopover. Reservations at the time, '21 as I remember, were not really necessary but couldn't hurt. Think I was in Site 31 as it was secluded and near water spigot. New Mexico is high on my list of places to visit next. Good Luck on your trip!