I was there right before Mardi Gras and used some of the restaurants in this thread: viewtopic.php?f=14&t=6089&p=190630&hilit=New+Orleans#p190630
I was well-fed while there, but the food was not up to par in any of the restaurants I went to from the above lists. I did walk around the French Quarter a lot, and there were quite a few restaurants that offered a vegan option. I didn't go into 13 Monaghan because it was a dark bar on a sunny afternoon, but it might be to your taste. My best bet was stocking up at the Whole Foods there - it was a really nice one.
I can't remember the name, but there was a veg-friendly restaurant in the middle of the Farmers Market in the French Quarter and they made my husband a very nice vegan lunch, special order, when he asked. Others we tried:
Cafe Abyssinia, ethiopian: I got a sampler that was fairly bland, but filling.
Lebanon's - really, really terrible overall, and after a long discussion with the waitress about what was vegan, my order came out with a big scoop of dairy yogurt in the middle; she suggested I just scrape it off.
Surrey's: the scramber my husband got was dripping in oil; my sandwich was good, but nothing I couldn't have made better.