I've been to the French Market a dozen times and here's the way I like to do it.
First get your beignets and coffee fix at Cafe du Monde. Then walk toward the market. As you enter the market proper, take a minute to orient yourself about what you are going to pass: drink, food, drink, food, trinkets, useless crap and wonderful people looking or trying to make a living. So about 50 yards into the market there will be a drink stand on the right. They feature a frozen pina colada in the upper left corner of their wall menu. Get one. Then start the rest of your walk. Frankly, this makes almost anything much, much better. So get you an alligator head or some handmade soap or some cheap t-shirts. There will be something that interests you; don't be a jerk, allow yourself to be interested. It costs you nothing and what else have you got to do anyway. And remember, keep sipping the colada.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.