As far as I know, all airlines that allow 13-year-olds to fly as unaccompanied minors charge for the service. It's usually only around $30, though, which isn't bad compared to the total ticket price. Southwest doesn't have a fee for unaccompanied minors, but kids 12 and over fly as adults, so you wouldn't be able to fly as an unaccompanied minor on Southwest. Most airlines don't require you to fly as a UM if you're over 12 (I think it's 14 for some, but usually just for international flights), so you would have the option to fly by yourself under the same terms as apply to adults, or to pay extra to fly as an unaccompanied minor.
I don't know how much you've traveled, but if you've flown at all before you might not want to fly as a UM. If you have a nonstop flight then it's not a big deal. The advantage is that your parents can go to the gate with you, but the disadvantage is that you have to get on the plane early, and on the other end you have to wait until everyone else gets of the plane before you can leave, so you're on the plane a lot longer. If you didn't fly as a UM and weren't sure where to go, you can always ask any uniformed airport employee and they'll be happy to help you find your gate (or even your seat once you're on the plane). There are plenty of adults who don't fly much and don't know where to go, so it's not like it's unique to being a kid or anything. It's mainly just a matter of if your parents want to pay to go with you to the gate.
If you end up with a connecting flight then it makes more of a difference. Besides having your parents come with you to the gate, you have an airline person take you to your connecting flight, so you don't have to deal with navigating the airport by yourself. But at many airports, you have to wait in this little room for your connect and you're not allowed to leave, which really suck. Also, if your flight is delayed and you don't make your connect, the airline will look after you until you can get another flight out, including putting you up in a hotel if there's not another flight that day. Personally I hated flying as a UM, but if you're not comfortable finding your way around an airport on your own it would probably be a good idea to fly as a UM if you can't get a nonstop flight.
I never noticed any difference between airlines for flying as an unaccompanied minor. The fees are pretty similar for most airlines, and they treat you about the same, so it's just a matter of which airline has the cheapest fare overall.
From New Orleans, I would use Southwest or Continental. If you have to go from Dallas/Ft. Worth, you might want to try American.
Try southwest airlines. They are really good about working with minors and the last I knew they did not charge anything extra for being a minor.