I agree with Millmoor. Elizabeth Goodale has nothing to do with your Paxton family in Dunbartonshire.
William Paxton and Elizabeth Goodale/Goodall had nine recorded children, all baptised in the parish of Dunbar, East Lothian
George, 1812
Ann, 1814
Elizabeth, 1817
Margaret, 1817
Andrew, 1822
Jean, 1825
John, 1827
William, 1829
Elizabeth, 1831
and a further child, Janet, baptised in Athelstaneford, East Lothian in 1835.
In the 1841 census at Burnside, Dirleton, East Lothian is a household consisting of
William Paxton, 45
Elizabeth Paxton, 45
Margaret Paxton, 20
Jean Paxton, 15
Janet Paxton, 5
all born in East Lothian
and in a neighbouring household are
Elizabeth White, 88
George Paxton, 25
Andrew Paxton, 16
John Paxton, 13
William Paxton, 11
Elizabeth Paxton, 10
all born in East Lothian.
Apart from Andrew, who would have been 18, these ages are exactly right for the family of William Paxton and Elizabeth Good*l*.
I also note the baptism of Elizabeth, daughter of Andrew Goodall and Elizabeth White, in Innerwick, East Lothian in 1790, so there is not much room to doubt that these are five of the nine children of William Paxton and Elizabeth Good*l*, living or staying with their maternal grandmother.
In 1851 Elizabeth Paxton, 20, born Dunbar, and her sister Janet Paxton, 16, born Athelstaneford, are living together in Spott, East Lothian. QED.