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As you alluded to, there are cruise lines other than Carnival that will accomodate 5 guests in a cabin but they are usually the family suites such as those on Royal Caribbean. Also, note that Carnival is restricted as to how many cabins on a particular cruise they can offer with 5 berths. The Coast Guard has rules about passenger loading. I think only certain ships can handle it too. The Fantasy-class ships can, and that might be a good choice for you since cruises on those ships also tend to be less expensive (older ships, but still very nice ships).
How to choose a cruise really depends on you. If you've never done it before then it is vital that you don't do this on your own by booking online or directly with the cruise line. Find an agent who specializes in cruises, and who preferably has CLIA credentials. You can find one near you here
The water on a ship is fine. No need to buy bottled water. I don't.
You can of course eat in the buffet, even for dinner, but the main dining rooms do a great job and they know that a family with kids doesn't want to linger. Go for early dinner seating and you'll probably be done in an hour or so. On our last Carnival cruise I think the average dinner time for us was 1 hour and 15 minutes (I keep track of these things). Or, you can opt for my time dining which has no set time. You can arrange a different time for each day if you want.