This may or may not help since I don't have a map handy. Coming south through MO, you may find a lot of zig-zagging on state highways. Ideally, you could get onto I-55 and just head south. When you get near West Memphis, AR, the traffic and construction is a nightmare....has been in various stages of construction for at least 25 years with no completion in site. Do your best to avoid it like the plague. Suggest you Mapquest it selecting "avoid construction" or similar choices. Also know, Memphis does not have the most hospitable drivers in the world. And you'll get no slack because you're towing. Don't think you'll have to deal with any toll roads.
Weatherwise....while y'all do snow in a big way, we do ice/sleet down here. Plan accordingly and know that southern drivers believe the faster you go, the sooner you'll get home. Or not. I would strongly suggest you not try for more than 300 miles per day due to constructions zones. I believe the construction just south of Memphis is cleared. If you're in need of a stopover in the Memphis area, suggest you check into the MS state parks....especially to see i they are open. Many close until around April 1.
Have a safe trip. Enjoy!