I live in Nashville and my wife went to Bama, so I have been to several Bama home games over the last 11 years. We've stayed in Tuscaloosa a few times, and we've also stayed in Birmingham a few times. Both options have their pros and cons, and I'd say it really just boils down to what type of gameday experience you're looking for.
Staying in Tuscaloosa: this is definitely the move if you're wanting to experience a full day of tailgating and then hit the bar scene after the game is over. I highly recommend this, because it's a blast. I would recommend getting there on Friday afternoon/evening and leaving on Sunday morning. Traffic in and out of Tuscaloosa is an absolute nightmare before and after the game, so it's best to just avoid that if you can.
Staying in Birmingham: Birmingham obviously has a lot more places to stay; so it's an easier trip to book. There are several charter shuttle services that run from Birmingham to Tuscaloosa a few hours before the game, and then they bring you back shortly after the game. There are several available, and I'm honestly not sure which one is the best, but if you google Tuscaloosa gameday shuttles you will find plenty of options. It's pretty convenient because you don't have to worry about finding parking, and you also don't have to worry about having a sober driver. Some of the bus services even provide food on the way there and the way back; and they'll let you drink during the trip. We did one of these a few years ago and it was pretty fun. The downside to this options is you miss out on the tailgating and the post-game bar scene. If you want to stay in Birmingham and enjoy the full gameday experience, the best move is to drive over early in the morning to avoid pregame traffic, then tailgate all day, and then hit the bars for an hour or two after the game and let the traffic die down before driving back to Birmingham. The key there is just making sure someone in your group will be sober enough to drive you back; which is definitely going to be tricky after we beat them 35-17.