For conceal carry, while in La, your Alabama permit will allow you to Conceal carry but you must follow La laws as they apply to CHPs while here. That means, no carry in bars. That may limit you right off the start. The carry in resturants which serve alcohol is a grey area and is basically agrued that if it's out of sight it's out of mind. State law forbids Open Carry in any place which sells alcohol for on site consumption. That means most resturants and all bars. Also, if I understand the law, your out of state permit will not apply to the Fed/State GFSZ rules. What I'm saying is that, from my understanding, if you get within 1000' of a school, your Alabama permit may not ve valid. <-I could be wrong on this.
Enjoy your visit but keep your wits. The crime problem in NO is real but you should be fine if you use common sense.