The cabs will likely be the only easy and cheap thing. Good luck finding a hotel room at an affordable rate asap! The hotel adjacent to the SuperDome is the Hyatt Regency and from what I hear is super nice with all of its renovations since Hurricane Katrina.

We stayed in the Holiday Inn by the Superdome one year when we came for a game and I would not recommend doing that. It is nothing about the hotels but there is not much around there that we wanted to do. We found ourselves either walking back and forth or taking cabs to the FQ and back. It made it difficult to make a pit stop at the hotel to drop things off or take a break.

I would highly recommend staying near or in the FQ and then taking a cab or walk to the game.

Most anything within the French Quarter or Central Business District (as typically designated) will be within walking distance of the Superdome. However, since you will only being going to the Dome once, and the NFL Experience is going to be in the Convention Center, I would think the ideal location would be within a triangle formed by the Superdome, Convention Center and Jackson Square (heart of the New Orleans experience), more specifically as close as possible to the intersection of St. Charles Ave. (Royal St.) and Canal St.