I can recommend both
RCCL and Celebrity 100%. I sailed on the Voyager which is similar to the Mariner. It was VERY kid friendly and had so, so much to do for them. My kids went ice skating, rock climbing, inline skating, played mini golf, etc. They actually enjoyed the kids clubs more on RCCL (although they went for 7 days on RCCL and 3 on
DCL so the comparison might not be fair). They do have casinos on the RCCL ships but its not that intrusive and if you don't want to be there then you don't have to. The passangers were very similar on all three lines. RCCL also has fun things like Johnny Rockets and an alternative restaurant.
Celebrity is really my favorite. They service and food on this line is outstanding. While you sit by the pool they serve you sorbet, they toast your voyage when you board with champagne, they serve tidbits in the casino, they have a social hostess that tries to make everyone happy, etc.e tc.
Both lines also have adults only areas. The kids club on celebrity is less jazzy and high tech than DCL but they generally have less kids sailing and the counselors get to know your kids well (again I am comparing a 7 night to a DCL 3 night so factor that in).
As far as price, a previous poster stated that a Cat 11 (inside) during march is about $5000. I am not sure if that is for four people but I am assuming it is. We sailed in a concierge stateroom with extra ammenities during March with three people for about $2600 (balcony room). This April we sail on Celebrity in a sky suite with a 145 square foot balcony and suite perks (butler, embarkation, in room dining) for two people at a price of $2500.