You might be interested in this recent thread: Passports, visas, etc. . .
Just show up in Freeport (or Green Turtle, or Bimini, or Georgetown, or Marsh Harbour) and say: "Hi, I'm from the US! Can I get a slip? Would you call customs and immigration for me?"
Hand over your passports and do some paperwork.
Then go back to the desk and say "Can I get a room, a shower, and a fruity drink?"
Then ask "Where do we get laundry done? Can I get a fruity drink near the laundry?"