Carnival can put 5 in a cabin (2 fold-down bunks, 2-under bunks and a roll-away) in some cabins. When we had 5 in a cabin, our kids were close in age to yours. The cabin was really cramped and I learned that two of my kids snore. We got an inside cabin, and I ordered flowers to look at so I wouldn't think about not having a window. We learned to keep the suitcases packed and under the beds, since the closets didn't have enough room for clothes for 5 people. I discovered the laundry on Carnival's ships, not that I wanted to spend a few hours inside a ship's laundry room. . . This was our first cruise and I was so excited that the crowded cabin was part of the adventure. (Now that my oldest is 15 and 6-6, we can't fit 5 in a cabin anymore.)
Some ships do have family suites, that can accomodate up to 6 people (Royal Caribbean, Disney). The cost of these suites are generally quite a bit higher, but the net cost might not be too bad if you get a lower rate for 3rd, 4th & 5th passengers in the cabin.
Some of the newer shps have connecting rooms. We had two inside-connecting rooms on Celebrity's Mercury, for the cost of a balcony cabin. We had to pay for 4 adults and 1 extra person in a cabin. We had looked at putting 4 in one cabin and taking an extra cabin with a single supplement, but for most cruises that wasn't a saving.
Your TA will be able to work with you to identify the least costly way of travelling -- and if you decide to get two cabins, will be able to book them next to each other or adjoining. Our first cruise hooked everyone in our family - your kids will love the kids' programs.