Just sailed Port Canaveral to St. Thomas two weeks ago aboard the Mariner. The time DOES change, you set your clocks ahead one hour for St. Thomas and the ship departs according to that time (St. Thomas time). And there are big reminders in the Compass. Kind of a hassle, but I set my digital watch ahead and we used that for "official time" while in the eastern Caribbean.
The funny part was when the captain very kindly assured us that, if we were real good, he'd give the hour back on Friday.
Maybe each ship is different?
On the Enchantment - sailed out of Fort Lauderdale - we were told to keep to the the ship's time & not Cozumel/Belize time, which was 1 hour earlier.
When the last tender left Belize at 5:30, it was 5:30 Enchantment/Fort L time ...