I used to live in NOLA!
Equest Farm (used to be City Park Stables) is right in the city and is a beautiful facility. It’s the only one I can really speak to, as I used to ride there.
Full care stall board (the only kind they offer) includes up to 12 hours of turnout and is $875/month. Pricey… but might be worth saving the cost of fuel driving across the Causeway every day!
Their website is http://www.equestfarm.com/
Miss that place
The other one I knew about was Cascade Stables, located in the heart of the city by Audubon Park. When I lived there 7 years ago I opted not to ride there because it was disorganized and a bit haphazard, but judging by their website they have renovated quite a bit, and the facilities look really nice. Stall board is $700/month, however it says that turnout is dependent on ring availability, so that might not be the best fit.
Good luck!