Ok, Darlene, here's what I've got.
In 1989 Philip H. Seaman published several works for the Gilbertsville Library. One of these documents was titled SHAW FAMILY and was basically a rough draft of the family history and a descendants chart/summary from those Shaw's residing in Butternuts Valley. It was from this document that I got Billy was the son of William.
After talking to Leah we know the following:
1. I did send her the Robert Darwin Shaw Estate documents and she has them in her file; part of which may be in what you have from them. Billy Shaw did have those 9 children with Sophia Dayton; vis. William Dayton (1811-1838), Elizabeth Heslop (1813-1880), Phoebe Dayton (1817-1884), Catherine Blair (1818-betw 1860-1870), Samuel Valentine (1820-1889), Darwin Lewis (1821-1894), Jane Anabella (1825-1893), Lucy (1827-1833), and John Jay (1830-1898). If you want a copy of the Memory Book for your sources you can write to Gilbertsville Library or myself.
2. The PHS documents have been updated since '89 and many of the family's he put together have inaccuracies in them with the new data...thus, Billy may, or may not, be the son of William and I'm to contact her in a couple months when her backlog gets diminished for an update. Now, you may have more updated info as you contacted her more recently than I. What is your source for Billy's father being Samuel?
I'd be happy to work with you and Leah to get this updated so the most current/reliable data is what's posted online.