There are 3 copies of JS in the Levy and 2 of them are dated 1842 by Ditson of Boston, both copies of the same by the looks. The 3rd one I take to be the Sinclair original published by Duff and Hodgson of 20 Oxford St, London, as it is described as 'A New Comic ballad'. My personal copy has the Duff and Hodgson stock number 1759 which Neighbour and Tyson date as circa 1853. All rather confusing, unless Sinclair wrote it in America before coming to Britain, rather like Sam Cowell.
Levy has a sequel 'Betty Sands' published in 1865, but this must be earlier as my copy advertises the sequel at the bottom of the pages.
Going by the style of the covers of the Ditson copies I would say they were much later than 1842. They resemble 1860s style to me. Perhaps their dating is an attempt at earliest publication. or when the song was written.

Looking at broadsides, it was printed in 1852 by the Glasgow Poet's Box. Bebbington of Manchester and Ryle of London both printed it in the 1850s, along with other later printers.
The Poet's Box gives the air as 'The Way to do it'.