The ships time will be set to whatever the local time is. So on the east coast, it will be EST, and when you sail through a time change, the ships clock will also change. Sorry I cant help with the ports youre going to but I believe all but Cozumel are on EST while Cozumel is CST???? Hopefully someone will reply, but you can also do a search too. Your daily newsletter on board will also remind you the night before to adjust your clocks.
Actually, I was looking thru some old Capers from our last cruise and in big letters it says on each port day PLEASE STAY ON SHIP TIME.
I think you should check with the cruiseline, but on Carnival, we've always stayed on ship time - which is the time zone in which you begin you cruise.
Carnival Elation March 11
Carnival Imagination Sept 07
Carniival Sensation Dec 06
RCI Sovereign of the Seas Sept 06
Carnival Miracle Sept 05
Carnival Glory Sept 04
Carnival Fantasy Jan 04