I suggest you head down I-81, then I-59 in Alabama to I-20, and then turn west.
My wife and I traveled cross country from Richmond, VA beginning the last week in February 2014. Even then, we were trying to stay in front of cold, bad weather. It was hit or miss. Fortunately, our second night out, we drove about an hour south of Birmingham, AL. We got a cold rain that night. If I had stopped where I originally planned, about 1.5 hr north of Birmingham, we would have been in sleet and freezing rain.
If I were you, I'd get to I-20 as soon as possible, but even then you could run into some bad weather.
