We are also a family of 6 and just returned from our 1st cruise! We loved it. We booked 2 connecting rooms - Cat 9's because I wanted the natural light but you can get Cat 11's much cheaper. We also did the 4 night cruise - to make sure we liked it (did not want to spend all that money and feel cooped up!) I was pleasantly surprised at the cost of our cruise, I was expecting to pay a lot more. Now I did book almost a year in advance so maybe that helped.
Keep looking and checking out different dates. Can your kids miss a week of school? We always go when the kids are in school and we just call it a life lesson! Different cultures ect.
Good luck and I hope you find a way to cruise with Disney.