Go to disney cruise line and click on reservations. Follow the instructions, enter the dates. If the ship is full it will not alllow you to continue. If thee is room, it will all the available categories for left for the cruise.
Actually, for DCL, Cruise Locators ( http://www.cruise-locator.com/ ) will show you only eight staterooms in whichever category you choose. So you don't really see how many staterooms are left in each category. If you change the number of passengers, you'll often see different stateroom numbers -- but, again, only eight of them.
As Werner Weiss stated, the cruise locators website will show you the next 8 cabins that match your criteria. If there were 28 cabins in that category available, you would not be able to tell the difference.
If you go to an online quote engine, you have to take into account that there might be different cabins available under different circumstances. For example, if the kids' programs are already at capacity and you enter 2 adults and one child aged 10, it would not have any availability. If you entered 3 adults for the same date, there might be 15 cabins available.
The only real way to tell how full the ship is for a particular cruise would be to ask DCL directly, and they would have no reason to divulge the exact info to a guest, so we'd probably never know for sure.
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