I take 45 from Mobile to visit family in your area often and it is the best way, boring as heck but the quickest. Once you cross Al state line it turns into two lane from the 4 lane but I have found traffic moves along quite nicely if there is any to speak of, and there are plenty of spots where you are allowed to pass if you do get stuck behind farmer brown, which in 3 years of my using this route I never have.
Then once you get to Mobile you can go one of two ways, One is around the west side of Mobile or the other is the through the city both are easy to navigate for the most part.
Enjoy your trip!!