You could plan to take I-95 down to I-40, across to Nashville on I-40 to pick up the Natchez Trace Parkway, then US-61 down to Louisiana. These Interstates are kept clear all winter, in winter weather country, although there are sections of I-40 that get caught by road blockage disasters from time to time, but these are long term when they happen, so you'll know not to go that way.
I can't guarantee the parkway will be open all the way through, it is not miantained as a critcal thoroughfare, so if it gets buried under ice, it could be closed for a while.
You will not get out of subfreezing weather so fast that way. Nashville and south might be above freezing, more like around or below freezing, and I've gone through in snow and freezing rain more than once in January. I-20 is a safer winter route, as bad weather is more often just rain, although it might be close to freezing.
But your other requirements, getting out of freezing weather as quickly as possible, suggests I-95, or even US-17, as a route down, because the closer you are to the coast, the more likely ocean air masses will have pushed away colder air coming off the continent. The years I lived in the Carolinas, it would be wet, cold, but above freezing on the coast when US-301 (where I-95 now runs) was covered with ice or slick with refrozen snow. I-95 should be better now than 301 was then, because winter maintenance standards are now better, but you can still be in freezing air until well south of Atlanta, January through February. I've also gone through the area in January with temperatures in the high 40s to mid 50s, something that seldom happens west of the Appalachians north of I-20 corridor.
I almost always have lots of alternatives in mind for these trips that go from one corner of a rectangle to another, and make my choices last minute based on weather, short term forecasts or what's happening right now. Often a diagonal looks shortest, but it is not always the best, there are usually more possible diagonals, and following the sides, straight south then west, or west then south, seldom costs more than 10-15% extra miles, and less often, extra time.
Tom Test
Itasca Spirit 29B
