John Paton and Lillias Duncan are my 6th great-grandparents.I am descended from their son David, Andrew's brother, who also lived in Shotts.
From the IGI, I believe Lillias Duncan was born 27 Apr 1769 in Largo, Fife, daughter of David Duncan and Isabel Kirkaldie.Lillias married John Paton 22 Jun 1788 in Dalgety, Fife.That's as far as I have traced it back.
I have your Andrew's christening date as 11 Jan 1805 in Shotts.
Hope this helps -- would love to share info with you.
P.S. "cottier" should probably be "collier", i.e., someone who works with coal.