I’m about a 10 or 12 hour-drive from New Orleans.
Train #19, the Crescent, goes through Birmingham. It starts all the way up at New York’s Penn Station and winds up in New Orleans. Might be a possibility for you sometime.
I guess that puts it out of your reach for a weekend trip but not so much if you want to do a vacation trip ... have you been to AMTRAK’s website? They may have something that would connect ....