Please give dates for birth, death, etc. William is a popular name.
You should contact the Mayflower Society to see if your Aunt submitted her lineage for membership at
http://www.themayflowersociety.com/welcome.htm and the local genealogical society where she lived as she probably turned in a pedigree chart or family group sheets. No one named Shore was listed on the passenger list so he would have descended through a female.
Another good source is
http://www.mayflowerhistory.com/