Sharon of BackTrack made some great suggestions in the comments to my last post.
I entered the first name and middle initial of my cousin, and the surname that my previous search turned up. I also selected California. Unfortunately, date of birth isn't one of the search fields.
The free results gave me the Name, Age, "Past and Current Locations" (city, state), and "Possible Relatives and Associates." I am also told what public records are available - 3 categories: Phone numbers, Addresses, and date of birth. In this case all three types were available, but I would have to pay $2.95. (Not a lot, but I thought I would try the other suggestion Sharon made first.)
I did make note of the possible relatives.
2) Back to Ancestry - and their California Birth Records.
Sharon suggested I enter the possible married surname for the surname of the child, and the mother's maiden name that I knew, and see if there was a hit. A great idea I should have thought of.
And there were two hits! And both children appear in the list of "possible relatives" at PrivateEye.
So if we are going to stick with the football metaphor, I have scored, which is impossible with a punt in American football. Perhaps I should have referred to this as a Hail Mary Pass. And since PrivateEye provided me with married names for the children in the "Possible Relatives" section, I've found another generation in the California Birth records.
Now to write the letter.