St Thomas will not stamp US PP
Cozumel will. Closest is the post office, one block north of the pier entrance, on main road, east side, metal colored roof(green or blue? ? ? , can be seen from docked ship on upper decks) Depending on who's behind the counter they may have different stamps. Tip is always helpful if taking a while to decide on stamp. Max cruise pax in line ever seen; 3. Fast and easy for us always. 100% success.
Grand Cayman. Often during business hours, weekdays, within the small secured fenced port area, on the south side, on right as you enter, is the immigration office, if open, has odd hours.
We have never been able to get stamps there though, because we are always rushing to excursion/beach/etc, therefore never tried in the am.
However the first two story building south of the fenced port is the alternate location, second floor. It's the third or fourth structure, like 500 feet max. Never have had a wait. Go inside, bank right to stairs, second floor on left. This is where we stop on the way back to port/ship for our stamps. 100% success.
Falmouth. No stamps available in secure port area. No idea if outside port. I keep telling them they should set up a both and charge for them. They are losing a potential revenue stream. lol
This info from fall 2014. 0 % success.
Castaway.
DCL will no longer touches PPs at CC. If you go the guest services and ask about PP stamps early, prior to docking, they may inform the custom officer, who may or may not be willing to stamp passports. Sometimes they don't bring the stamp. Sometimes the official refuses. You have to wait at guest service with all the PP you want stamped. One person can wait with "your cabin's/family's PP. If CM's can convince the official to stamp PP and they have the stamp, they will come up and stamp them in front of you. We have been 50% successful getting stamps at CC when we have docked and woken up early enough. Pax waiting has varied; 2 -10, of course with more PPs than that to stamp. 50% success.
HTH