Shame you're not earlier in crawfish season. Try some oysters.
Avoid oysters. They're out of season.
Check out Bachennal in Bywater. Michelin star tier food served on paper plates in a backyard behind a wine store. Buy a bottle of wine on your way in. It's one of my favorite places but is kind of lamestream now since it was featured on Treme.
Another cool place is The Country Club in the Marigny. It's a restaurant/bar in an old house with a no shit clothing optional pool out back.
Don't forget to eat beignets at Cafe du Monde in the Quarter. It's literally the most touristy thing to do, but you've never been, so you have to do it.
Edit: Another French Quarter recommendation is EAT. I seem to go every time I'm visiting because it's one of the more authentic-feeling places (not right on Bourbon) in the Quarter which is generally beautiful and a fun place to be. It's BYOB so grab a bottle of wine before you go and they'll pour it for you and the food is always good.