Hello Becky. I found your message and we may have a connection here, I have a Willie J. Bradshaw whom had a son John O. Bradshaw. I can not say for sure at this time if Willie was short for William or not. But I found it interesting that your William you say was the father of John O. I recently visited Arkansas, and found John O. Bradshaw buried at Hardin Cemetary along with John W. Bradshaw beside him.
The Willie J. Bradshaw in my line was the brother to my G. Grandfather George S Bradshaw. I myself am at a huge brick wall in my research with my family. Please get in touch with me, I will call my G. Aunt and see if she knows if Willie stood for William? By the way my Bradshaw line were mostly all from Arkansas/Greene-Clay county area.
You can e-mail me at firstname.lastname@example.org
I will look forward to your response.