I would look for any Cain family on the 1900 census in Franklin Co, TX.Even if it was taken prior to William's birth, his should be one of the Cain families living in Franklin Co, just prior to his birth.Then I'd check just those Cain families on the 1910 census, and you'll probably find one of them has a William that fits the right age.This would allow the fact the Cains mayhave moved between 1900 and 1910, and will eliminate all the other Cain families on the 1910 census. (^_^)Also, you will pick up any other siblings in the family.Maybe you can track one of them easier then you can William.And, there's always the chance descendants of William's siblings know something that can help you.
Have you talked to members of your father's family?His mother may have been bitter and not talked of his father, but sometimes family members know information and are willing to talk about it.Even cousins may have heard family stories.
You have a birth date, you know where he registered for the draft, do you have a Soc Sec number for William?You willneed a little something to be able to prove who he was, but your local VA Office may be able to request his VA records.I don't know if they will do it if you do not have a death certificate.They will expect some info.Myabe a copy of your father's birth certificate would help.That VA record may tell if William ever married again, or had any other children.If the VA Office can request the record, you will have to go in and look at the file, then request which pages you want copied, pay for them, and they will mail them out to you later.
If William is listed as the father on your father's birth certificate, I wouldn't say anything about a question if William and Edith were ever married.