Although interstate sales have to go through an FFL, I believe that it is legal for someone to ship (not mail) a firearm to himself in another state, care of whoever he is visiting there. For instance, if live in one state and are going hunting in a distant state, you can ship your rifle to yourself, care of the person with whom you are going hunting, in the other state.
There ought to be a UPS office outside the city limits, in a suburb where it is legal to own a handgun (and presuming you have an Illinois Firearms Owner ID Card), that would accept it for shipment it to you, in Florida or in care of someone in Florida. You fly out of O'Hare, the handgun flies UPS, you meet in Florida, and live happily ever after.
Golly I'm glad I don't live in Crook County anymore...