I can pack for each of my family in half a carry-on sized bag. I think we took 4 bags (22x9x13) on our last cruise. The 4th was mostly pee pads (we had a leaker at that point and I didn't want to deal with calling house keeping for a linen change in the middle of the night) a puddle jumper that we didn't end up needing, diapers, and other stuff to make up for not having washer/dryer access. This was RCI to Alaska, so we needed more layers than we'll need for the Caribbean, especially since
DCL has guest laundry access.
Each person gets clothes for 4-5 days regardless of how long the trip is. I do try to pack enough underwear for every day of the trip.
Typically, the warmest outfit is worn on travel days. Everyone gets a full change of clothes in their personal item. I also tend to put another change of clothes for every one in the carry-on.. So really I'm only packing 3-4 outfits/person in the checked bags. Honestly, I tend to either do carry-on only, or avoid a carry-on altogether. My older kids are old enough to manage a roller bag now, but when my kids were really little, I pretty much only traveled with a backpack as a personal item. Each of my kids has carried their own personal item, since they were 3 or 4. Typically, it only has a change of clothes, and a few toys in it. I'm pretty harsh with my kids about not bringing a lot of extra stuffies and such.
With kids that young, I would bring a small umbrella stroller....or 2, unless you're on one of the smaller ships. I would not bring a large double stroller, because the ship hallways are just too narrow.
I'd also consider picking up diapers in your departure town, or requesting them from the ship in advance rather than bringing them from home. Paying a little extra for convenience is probably worth it, especially since checked bags can get lost.
I also tend to pack a packable duffel for extra space on the trip home. But, I limit my kids souvies pretty heavily too, in order to minimize packing issues.