We are huge Disney fans and we will always pick them first if the opportunity it there. However, we did take a mediterranean cruise two years ago on Celebrity and we were able to book two balcony cabins for 1/4 of what we would have had to pay for a single cabin for 5. I would definitely recommend this route without kids. Out of 12 days, only two were sea days. We were at port all other days and we had tours booked for each port. So as someone else noted, it is more like a floating hotel. When we got back on the ship, we would eat, maybe watch a show and go to bed so we could be rested for our next adventure. If you were traveling with your kids, I might recommend Disney but only if your kids are younger. Mine were 10, 12 & 14. While we wish they had more offerings for the kids, I still can't justify the $10,000 premium (though I might for the Norwegian Fjords or another "specialty cruise"). The money you save will get you some pretty awesome tours.
I know you said you don't really want to be gone for more than a week because of babysitting issues and I totally get that. However, I second Offin's recommendation. Ours was 12 days and the travel time and time zone changes took awhile to adjust. And there is so much to do and see that anything less would really feel too short. Of course, I would not pass up a shorter cruise if that was my only option.
Incidentally, we visited Greece (Athens, Mykonos & Santorini), Italy (Rome & Naples) and Turkey (Istanbul & Kusadasi (Ephesus)). I was not super excited about Turkey but we really enjoyed both ports and those were two of our best tours. We loved our cruise and I really can't wait to go back to Europe. I feel like I learned so much.