You are welcome to use the companion restrooms located throughout the parks. They are just next to the regular restrooms. They are usually in between the men's and women's rooms. There are usually one or two of them, and the door just has both a man and a woman, and usually a handicap logo and the word "companion", but it is NOT JUST for handicapped or disabled people. We use these types of restrooms as a family of 4 at
Disneyland resort often. (me, DH, and two toddler boys. We do this ONLY when there is no one who is waiting for it already who is obviously disabled and needs it, but our son has autism, and rather than trying to deal with him in a TINY stall in the regular restroom, as well as trying to manage a "potty training" 3 year old, we use the companion restrooms since they are nice and large, and allow us to manage the kids better (and actually faster).
I don't know how the baby care centers work at WDW, but at Disneyland Resort, they will only allow children UNDER 42" use the restrooms in there, because they are "small potties".