I've traced back to a William Shaw a Seaman in Greenock who married a Mary McGee on 10th June 1816 but here I'm stumped as I don't know his DoB and haven't had access to look at the 1841 census.
They had a Martha (b.1816), James(b.1819), William(b.1821 d.1901) and Joseph(b.1824).
William, the son, a Ship Carpenter married Catherine Sinclair in Greenock in 1844.
I wonder if my William Shaw the Seaman could be your William born 1801? Son of John and Elizabeth.