Re the late, great Frank Taplin ("Q")'s report on "Old Moll Roe" above (2003), Alan Lomax recorded a stanza and refrain under the title "I Met Miss Monroe in the Morning" from a lumberjack at Round Lake, Mich., in 1938:

https://www.loc.gov/item/afc1939007_afs02322b/

The tune he uses is neither "Rosin the Beau" or "Tarpaulin Jacket." It's closer, in fact, to that associated with "Four Old Whores from Baltimore."

Rudyard Kipling, in The Light That Failed (1891):

'What shall I sing?' said [Dick], turning in the chair.
'Moll Roe in the morning,' said Torpenhow, at a venture.
'No,' said Dick, sharply....The old chanty whereof he, among a very few, possessed all the words, was not a pretty one, but Dick had heard it many times before, without wincing."

Kipling also wrote (in a letter,) "“The noble tune of ‘I met Moll Roe in the morning’. The original words thereof are long-shore bawd pure and simple but the air is a cyclone as I once heard it. There’s a hymn tune that fits it as well and there are about twenty tunes to Moll Roe."

It must have been widely known among men. There's a stanza in the American "Immortalia" (1927).