The song is by Dibdin, also known as Jack The Guinea-Pig. You can find a broadside copy at the Bodleian and online copies of the text at archive.org. None of these have tunes.

The National Library of Ireland has a sheet music copy with piano acc: A Sailor's Life At Sea (a favourite song as sung at Sadler's Wells with applause), but I can't find an online copy of the music.

Mick