I wonder if the occasional attribution to Dibdin (presumably, the elder) is a mistake. He included it in his collection Sea Songs and Ballads, which contained both his own songs and songs by other people. But I haven't found it in the collections of his own songs which were compiled by his sons Thomas and Charles jr. In fact, the title Dibdin gave in his own compilation was "Still from Care and Thinking Free". The Busy Bee, Vol. 2, says "Sung at Sadler Wells by Mrs. Harlowe" and "Written by Mr. Langdale." Another source gives a performance date of 1790 for "Jack the Guinea-Pig" and attributed it (or "A Sailor's Life at Sea") to the composer Mr Taylor.

Whatever the case, I haven't been able to turn up music.