New Orleans is a fascinating place, but isn't really a beach town; it's more of a river city and port town, with it's own unique flavor.
St. Petersburg, FL has Fort DeSoto, with lovely beaches and camping on or near the gulf. It's probably best done if you have a toad or tow vehicle and don't need to drive an RV everywhere, assuming you want to do things other than waterfront activities. Reservations over the weekends for next month are probably non-existent at this point (or practically so), but it looks like some sites are available during weekdays.
There are lots and lots of other great options, of course, some of which have been mentioned already.