Off season is good, however, the Disney cruises are almost ALWAYS completely booked. I would suggest a warmer season, but not Spring Break or early summer. You may also want to avoid Thanksgiving week and the Christmas holidays.
We REALLY enjoyed our balcony-- coffee in the mornings, drinks at night, standing outside when docking and leaving port-- however, it is quite a bit more expensive. With young ones, I would definitely get a window or view room so they can see outside.
Check out the kids' space as soon as you get onboard. Do NOT feel guilty about taking them there. They have a BLAST so the sooner they get there, the sooner they can have fun.
We liked getting the late dinner seating, so you have more time to see movies and stuff but if your kids are 'early to bed' you may want the earlier seating for dinners.
Bring some sort of magnet or small magnetic whiteboard for your door. All of the doors tend to look alike and having 'something' on your door helps you and your kids find your room easier. You can also leave notes for the rest of your family (husband) about where you are at any given point. Cell phones do NOT work on board but they give you a 'walkie-talkie' kind of phone in your room that WILL work onboard.
Each evening you will get a 'magazine' called The Navigator which lists all of the activities for the next day. We always took ours to dinner and discussed what things we felt we 'had' to do and which ones we felt like skipping in lieu of just chilling.