Basically, when cruises haven't sold enough Disney will release rooms at discounted rates. Usually this happens once it's inside the 75 day paid in full date so that people who booked ahead will already be paid & subject to cancellation fees. Some special sailings will offer discount rates before the 75 day mark but that's not very common. You just have to look. The way to find these discounted fares is to go into DCL's website searching for cruises. If there's discounts available you will see them listed as a stateroom category at the top of each type of stateroom tab. IGT stands for inside guarantee, OGT for ocean view guarantee, and VGT for verandah guarantee. When booking these discount cruise fares you do not pick a stateroom number/location. One will be assigned to you usually within 10 days of sailing. Each of these guarantees you at least a room of that type, not less. So you won't book a VGT and get assigned an inside stateroom. You'll get at least a verandah room. But, if you book an IGT it's possible you could get assigned a higher category like an ocean view or verandah. It happens from time to time. One rule to remember with these discount fares is that you MUST pay in full at the time of booking and I think all monies are 100% non-refundable if you cancel.
Right now *GT fares are on most sailings thru the end of July and into early August here & there. I'm watching carefully for late September/early October.
Because of your family size you may not find a whole lot pop up. Try looking with just 2 or 3 of you and get 2 rooms. I'm not sure how it would work getting 2 rooms and guaranteeing they are connected. That may not happen. Maybe call Disney Cruise Line and ask on that one.