We are a family of 5. We love
DCL, and will pay extra in the Carib. for DCL, but don't find it worth it for port-intensive Med. We have done Carnival, NCL,
RCCL, Princess, plus DCL. RCCL has a few OV cabins that sleep 6 persons. You have to have 5 to book them and you have to call. If you can get one (they go fast), it's a god value. NCL Epic has rooms for 4 plus studio rooms, so if you split up and DH sleeps in the studio, it's a great value also. In high season, 2 years ago, 5 persons in the family cabin was about $7.5K, Epic with 4 persons cabin and studio $6.5K.
For Europe. We like RCCL 2nd to DCL. RCCL has a good mix on entertainment, kids clubs, decent food, good pools, etc. EPIC had better MDR food and some interesting shows, like Blue Man group, but the pool area was VERY congested--like stepping over people to get by. Princess in Europe was somewhat boring for us (40s) and I didn't see many kids when we went without ours in early Aug. We like Princess in the Carib. Probably #2 after DCL for us. Carnival was almost 10 years ago as a couple. It was a new ship, just built in Europe. We liked it a lot. There are some reviews online for the Vista, which sailed this summer in Europe. We won't sail Carnival in the US, based upon reviews (and having sailed 2X), but would not hesitate to book Carnival in Europe.
Unlike Carib., the ship/entertainment really doesn't matter as much, esp. in the summer in Europe. It's hot and you are touring around most of the day. Coming back, grabbing some quick food, hop in the pool, relax for an hour or so, dinner and then maybe a show, but many times we just watched a bit of TV in the room and got ready for the next day.
RCCL kids clubs for under 12 are fine. DS9 even opted to stay in the kids club vs. a 1/2 day tour. On port days, kids clubs are open all day. At sea 9-11, 2-5, 7-10ish are the hours. This worked fine for us. RCCL also has some ships with active things, like flow rider, rock wall, ice skating. If that interests your boys, then I would choose RCCL. With 5 persons, RCCL is always my 1st phone call b/c of the cost savings if I can book one of the above rooms.