I live in New Orleans also and there are plenty of places to hunt if your willing to drive. If you want somewhere close then there is Pearl River Management area. The best places you will need a boat but you can walk in several places. Most walkin places will be packed as there just aren't many places and most people are lazy.
Next is Sherbourne which is about a 2 hour drive west of I-10. This place has plenty of big deer and you just need to go deeper in the woods than everyone else. About a 2 1/2 hour drive from NO is Thistlewaite WMA which has your best chance at a monster as they have antle restrictions in place which is on its 4th year of a 10 year plan. Visit the wildlife and fisheries website as posted above and you can download maps and directions. The secret to public land hunting is to get away from the crowds and be willing to put in the work. You are a marine so you should be fit enough to go deep in the woods. This is key and be willing to hunt all day. Move around some if get bored with one area.