You are guaranteed legal to travel with the firearm because of the federal Firearm Owners Protection Act that was passed a few years ago. You are constructively legal to carry the firearm out of the airport and secure it in a vehicle as proscribed by law (unloaded, cased, ammo separated) although if the wrong LEO is made aware of it you could get a real bad deal until you are finally vindicated (think jail, attorney $$$, etc.) If you do this, act like the legal upstanding citizen you are, and make sure NOBODY knows what you are doing that isn't required to know by law (TSA, airport check-in counter).
Alternatively, a local FFL could mail the weapon to a FFL near your final destination and you get to drive down unarmed.
Finally, I'm sure you'll get more eloquent and particular advice than I am capable of from others shortly.