1870s is really the earliest decade about which the John Shorts and Carpenter's singers and people like that could tell us, I think -- Sure, a few were at see in the 60s, but how does one reasonably separate what one may have learned in the 60s from the 70s? And since my division by decade is really somewhat arbitrary, "late 60s" is not necessarily a different "era" from, say, 1974.

With Carpenter's informants, at any rate, I think that we may safely set the parameters back 20 years before your cut-off date. Many of them were at sea in the 1850's and some even in the late 1840's: it was not unusual to have 12-year-old lads on board.
* Edward Robinson - born 1834 - to sea 1846
* Mark Page - born 1835 - to sea 1849
* James Forman - born 1844 - to sea 1856. Forman claimed that Bully in the Alley was "learned as a boy before going to sea", and also gave Carpenter a song "learned when seven or eight years old; sung in 'guysens' (serenading, with faces blacked) on New Year's Day; did not celebrate Christmas"