If you ask for Family Restrooms, you will likely be told they do not have them. They DO have Companion Restrooms, which are restrooms set up as a single person restroom with a lockable door. They are set up to be accessible for handicapped use, for someone who needs assistance in the restroom or for situations like yours where the regular restroom is not a good option.
There are only a few of them in each park, which is why they do not call them Family Restrooms - if they did, they would always be full of families and people who could not use the regular bathroom would always find them full.
They are listed in the Disneyland park access guide for guests with disabilities as:
First Aid -Golden State
Golden State regular restrooms
Flik's Fun Fair
At DL Park
First Aid
Tom Sawyer's Island
Rancho del Zocolo
On the right side of the page, there are links to PDF files with specific information - the one for guests with wheelchairs has the Companion Restroom info.
Actually in California there are called Family Restrooms - labled just that way - and they are sprinkled throughout the park. There is one I specifally remember in the bugs life land.
Thank y'all so much, this really helps. I think we used the restroom at Bug's Life. We also found a good bathroom in the Carnation Plaza area when we visited last summer. Good to know of some more.