Hi Amy,
I have a couple of ideas to give you the space you need yet keep the cost more affordable. My first thought is the family suite at the
All Star Music Resort, The family suites are made up of two rooms. One room is a bedroom with a queen bed and bathroom; the other room is a family room with furniture that converts to sleepers to allow four more spots for sleeping. It also has an additional bathroom and small kitchenette (basically a small frig and microwave). Technically, the room sleeps 6, but four of your little ones are under the age of 3, so Disney reservations would need to confirm if this is permitted.
Similarly, you could book two connecting rooms at any of the
value resorts, which would allow 8 guests. You'd also have 4 beds, and you can request bed rails if your little ones are sleeping with those (I can't guarantee their availability, however). While I don't know the specifics dates you are planning on visiting, the first of the year can also bring lower rates making it possible that two connecting rooms at a
moderate resort may also be a possibility, which would give you bigger rooms. The current special offers at WDW can be found
here.
Please let us know if we can help again as you continue your planning, and thank you for visiting the Moms Panel!