A few thoughts:
I was born and raised in New Orleans for 30 years. It is the most incredible city I've been to and is certainly a blast for kids. IF you look broader than Bourbon.
1. Paddle boat cruise from FQ to Audubon Park for the Zoo. There is an Aquarium/Zoo/Cruise package.
2. FQ and French Market are a lot of fun in the daytime and relatively safe. Start your morning at sunrise having beignets at Cafe' Du Monde and spend the day walking the streets. You can spend hours in Jackson Square alone watching the street performers.
3. Cemetery and Ghost tours
4. Take the streetcar from the FQ all the way down St. Charles and have breakfast at Camellia Grill
5. Go out of the city for a canoe tour of the Atchafalaya Basin. Massive Cypress trees, alligator and the cajun culture all around.
6. Do a charter fishing trip out of Plaquemines Parish (I know people) to fish for redfish in the shallow water marsh. I grew up doing this, and there are few other fishing experiences that match it.
I could go on and on...
The Alabama coast: They have very nice beaches, but not that much else. I would be disappointed having driven 1k miles.
I also lived in the above mentioned town of Santa Rosa Beach (actually I lived in Blue Mountain). There isn't much here but the beach either, however, it has even nicer beaches (IMO, the best). There are a few planned communities that are actually a lot of fun to spend a day at. Look up the website for "Seaside, Fl". In a prior life I was a Landscape Architect and had a very small hand in planning parts of this community. You can rent bikes, go to the beach market, nightly outdoor concerts. Nearby is the town of Destin or Panama City Beach. Both have all of the typical beach town family things...Putt Putt, water parks, etc. Not a great destination from that far, but better than Alabama.
Congaree: We live about 2 hours from there now. It is a very cool, small park. However, you've seen it in about 3-4 hours. Maybe tie that as part of a larger loop with other stops? For instance, Outer Banks, Congaree, Charleston, Hunting Island (depending on storm damage), Savannah. There are some interesting black water canoe trips through the coastal areas of SC (Edisto River), but we've never done them.
I think the flavor of any of these southeast trips is going to be slower paced and more relaxing than a run and gun, see the sites kind of trip. Just be aware of that. You might have to work at keeping the kids excited.