LSU and Tulane do not have any kind of "combined program." The medicine services are run totally separately. This is the case for all programs at the 2 schools: They both handle patient care in the Charity system. For example, on medicine service (though this holds for Ob/Gyn and probably peds and surgery), they split the call nights. Therefore, your team is on-call q4 (which is totally in line with what programs around the country have for their call schedules) and your service picks up all new patients taken in that night. If there is a former LSU patient being admitted on a T night, the LSU team on-call will pick up that patient. Otherwise, the Tulane team gets him. The opposite is true as well. So, on any given night, both teams are on call manning the Charity and University ERs. They rarely interact with one another, much less compete or get in each other's way. Also, students work with staff and house officers from their school only. Therefore, LSU students are not competing with Tulane students for attention or vice versa. Once a month or so they have a combined Grand Rounds, to share some intellectual banter, but otherwise they couldn't be more separate if they were in separate hospitals. I wouldn't say Tulane runs the show either. It took them until this year to even get their name on the sign outside the building, if that tells you how much they care about Tulane's input.
As for which school is preferable, I would say that you will probably get a comparable education, but UAB has a stronger reputation. That might provide you more with more options when it comes time for residency. Also, I have heard that their physical plant is much nicer than LSUs. The LSU classrooms are very nice, but the hospitals are old and dirty. It's an amazing experience and I don't necessarily think you should decide how much a hospital has to teach you based on how new and high-tech it is, but if that's what you're looking for, go to Bama. Also, both are state schools, which, in my experience, tend to be somewhat cutthroat. I have heard this about both schools, so I don't know how to separate the 2 on that point.
Lastly, I could be mistaken, but I can't think of another level 1 trauma center in the New Orleans area. Every trauma comes to Charity, period. Plus, the pathology in the New Orleans population is second to none. If you go to LSU-NO, you can be guaranteed to work in NO for a good portion of third year (with a rotation here and there in Baton Rouge or one of the other Charity Hospitals around the state), instead of being shipped to Tuscaloosa or somewhere else like at UAB.
Good luck on