It's certainly not the best time of year for it but it's perfectly doable. You will probably have some cold weather here and there but it shouldn't be real bad - maybe dipping to freezing in the morning a time or two. The only cold weather I had was just below freezing in the early morning in Mobile. Other than that cold wasn't a problem.
The best roads you'll have are in Texas and the worst are in Louisiana.
Don't hesitate to ride the shoulders of the interstates out west if it's convenient for you.