It's so funny to hear your experience at Grand Cayman, Linda! We asked where to go and someone pointed us to the Immigration office. So, we go up the steps and the door is locked. We stood there for a minute, not sure what to do. We had just turned around, when a guy in an official looking shirt walks up behind us, says "You want to go in?" We said yes. He then opens the door, goes in and closes it in our face as we're about to follow him in. Another lady walks up behind us to go into a different office. We asked her what was going on and she says "Oh, he doesn't work there", then goes in her office and shuts the door. What??? The first guy comes out and pushes past us, so we knock on the still-locked door. Finally someone answers and tells us that they don't want to stamp any passports now and that we have to come back. We came back later that afternoon and the door is now unlocked. We go in and the man inside tells us that the Customs officer has left for the day and we're out of luck. Seriously?? So, no Grand Cayman stamp for us.
At Costa Maya, we had an excursion, so we didn't even try. In Cozumel, we asked someone and they said it was a holiday, so no government offices would be open, so don't even bother trying.
4 cruises and we still have no stamps on our passports... Guess that means we'll have to keep trying!