First and foremost, welcome to New Orleans! These red flags certainly are cause for concern. As Ian pointed out, the sponginess and wood damage definietly sounds like termites. In Louisiana every lender will require a clear wood destroying insect report prior to lending money to purchase a home.
There are a few other reasons that the house may not be level. You didn't mention in what part of New Orleans this home is located. Some areas (Lakeview , Gentilly, and New Orleans East for example) many homes experience a foundation shift as these area required a lot of fill when they were developed years ago. Smart homeowners have loads of dirt delivered every couple of years to shore up the foundations in which case, the cost is minimal. If the foundation has shifted severely, there are shoring companies that will drive a more permanent foundation (pilings) underneatht teh existing structure. This can be quite costly, but will salvage the home. The reputable companies offer a lifetime warranty. Other homes are not level due to shoddy workmanship in which case you have a serious problem on your hands that may not be remedied through shoring.
If you are dead set on this particular home, your best bet is to hire a licensed home inspector who has experience with foundation issues. Your buyer's agent should be able to refer you to a reputable inspector and shoring company. If he cannot do so, don't hesitate to contact me.