Many songs explore similar themes. "Rocking the Cradle" possibly followed Irish (and British Isles emigrants?) to corners of the British Empire, but it has not been reported from the Caribbean, nor from the U.S.
The song sheet may or may not have been common; related variants are likely uncommon since it is lacking in rather extensive U.S. collections of UK and Irish song.

"The Unfortunate Rake (St. James Hospital, Bad Girl etc.) seems to me to be unrelated to Caribbean songs; more likely the jazz version of St. James Infirmary was picked up there.