Panzer,
I made pretty much the same trip you are looking at last year. Drove from Florida and went to Fort Morgan and Fort Gaines at Mobile Bay. Visited the USS Alabama and museum (well worth it!).
On to New Orleans and visited Chalmette Battlefield (battle of New Orleans). Be careful trying to find the entrance though, it's in an industrial area and you have to cross the rail road tracks on a gravel road to get into the park. We passed it but the first time.
Take a walk down Jacksons gun line and fortifications, there's nice markers with information. Walk to the River to get a better picture of the position. Then drive around back to the British positions. It's not going to take up a lot of your time, and the museum is nothing special either. The monument is really neat though.
Fort Jackson may not have much to offer as a lot was damaged during hurricane Katrina and a lot of exhibits damaged. Did not go so you might get more info.
The WWII Museum is fantastic and will take a full day. They were adding a new section last year so it might be completed now. We did take the hop on hop off tour around the city that was well worth it as well and taking time to go to Bourbon St and the French Quarter. If you don't have a hotel yet, I highly recommend the Brakeman Hotel!
From there we drove to Vicksburg and spent the night then a full day at the battlefield. Lots to see. Down load the APP and do the auto tour is great. There is a separate museum on the tour just for the USS Cairo which is on display outside the entrance of the museum. The Vicksburg visitors center is very good and has lots of cannon outside including a not often seen 14# type II James rifle as well. Along the tour you will see more 30# Parrott rifles then you have ever seen!
A trip to downtown Vicksburg should be worthwhile. Also, you should take note on the tour as to the changes in course of the Mississippi river has changed since 1863.
As for other battlefields, Champions Hill is east within a short drive, but the its all on private land and not marked very well.
Make sure you visit the southern half of the Vicksburg battlefield (south of Clay Street).
Hope you have a great trip!
Kim