If you're intent on taking the meat back with you, I suspect Texas is your best bet (never hunted there...so I could be wrong), but for a true free range spot and stalk axis hunt, I'd suggest Hawaii....specifically the big island.
I am in the process of getting a trip set up there for pigs and axis myself. Getting the meat back would be more of a headache, though the outfitters there have played this game so much they got it down. Flash freeze it and overnight it. It'll cost more to do so, but the hunt is probably more what you're looking for.
