You should really check out the prices for MSC lines in Europe. They're sort of the Carnival equivalent in Europe (flashy ships, okay food, lots of families, a bit of a party atmosphere on the pool decks) and they have lots of kids sail free promotions so you can get amazing rates. We sailed a seven day Eastern Med with them in 2014 for $4500 Cdn for 2A2C and I'm booked on a seven day Baltic for Aug 2107 for just under $5000 Cdn in an extended balcony. (I think it was $3800 US before the conversion)! We were a bit worried about trying something new, but if you mainly want to travel for the ports its an amazing deal. The rooms were really nice (Maybe even a bit larger than
DCL) the food was fine, service was good, the entertainment was pretty lame and the kids clubs were nowhere near as good as Disney but we saw six amazing ports in seven days for an amazing price and had a great holiday.
MSC tends to fly under the radar a bit since they mainly market to Europeans and many Americans don't like the feeling of being in the minority on a cruise ship. It did get a bit tedious to have to listen to every announcement five times (Italian, English, French, Spanish and German) and the my kids felt a little overwhelmed with all the different languages in the clubs but their prices are often half the price of the North American lines, which more than makes up for having to deal with five languages and a bit of culture shock!
