All drinking water does come from the desalination units on board the ship. Most ships do have fresh water holding tanks that can be filled in ports but that is not potable water for drinking purposes and can be used for washing down the decks and other maintenance purposes. On board a cruise ship all water that comes out of a faucet that a passenger uses comes from desalinated water.
BTW: The water in the toilets is salt water. Don't ask how I know.
Mike (An engineer told me.)
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"There is a great difference between being well traveled and just having been to many places." ~Me