Hi Lisa!
I did a little research on this topic and I have come up with several options for you. First, most people recommend using the First Aid Station located within each park to change the diapers of either handicapped children or adults. The First Aid stations will have private rooms with cots to use if he needs to lie down in the process.
Another option is the Baby Care Facility that you mentioned. There are no private rooms here, only one large room with multiple changing tables. I'm not sure that this would be the ideal situation for you. However, there is a companion restroom located in each of the Baby Care facilities and one is indeed located in every park.
Finally, if your son is able to stand on his own, many companion restrooms are located within each park. I found a great
website that has photos of every restroom in Walt Disney World, and notes those with companion or handicapped stalls. Simply click on the "photos" icon and then select "WDW restrooms."
Be sure to check with the Cast Members once you arrive at WDW. I find that the folks at Disney World will go out of their way to make sure that guests needs are met and that everyone's trip is magical! Have a great time with your son!
Much pixie dust,
Jackie S.