Oh, yes, I can certainly see where you're confused. Note: I have tried to contact Gilbertsville Library and Leah's not in...I'll try again later.
OK...this is a lot so bear with me. I've been researching this for nigh on 20 years and have accumulated A LOT...
First, in August of 2002 I received information from Susan Shaw Jinn regarding documents from the Estate of Robert Darwin Shaw (who was the son of William Sage Shaw, who was the son of Darwin Lewis Shaw, who was the son of Billy Shaw). There were 44 documents in this information, the last being photocopies of Billy Shaw's Memory Book. This packet is where much of my info comes from for Billy's children.
Second, regarding Billy's parentage...i took my information from "The Shaw Family" a document I received from the Gilbertsville Libary in 1998. I did a skim through the other histories I have and I cannot find another supporting document. I have lots on Billy, and some on the Shaw brothers who settled G, but nothing linking them. I will try a more detail search later and post, but my 2-yr-old twins are getting into things and I need to go for now.
If you'd like copies of those 44 documents, please send me a private email at namccarty @ earthlink.net and I'll just make double (one for you and one for the library if they don't have it) asap.