For any interested Shaws out there--
I found the connection in a book called "The Edward Clarence Plummer History of Bath Maine" 1936, Bath, Maine.
It mentions the relocation of Elisha and Susannah Shaw to Bath on page 97:
"Elisha, founder of the Shaw family [in Bath, ME], arrived about 1761, building a log house on the east side of Town Road just above where Pine St now intersects. His good lady, Susannah, one day shot a bear which she caught carrying off one of their sheep, an exploit which has given her name a place in local history."
David Turner Shaw, the youngest child of Elisha Shaw, was born in Bath, ME. All of his other siblings were born in Massachusetts. There is quite a lot on David and his brothers Joshua and Elisha Jr. in Bath history.