The song turns up in a number of forms, of course, and with a number of different titles. The Fireship as already mentioned is evidently not quite the same version as the one Paola is looking for, but without knowing where [s]he heard it, we are really just guessing at the moment; unless, that is, somebody has recognised the precise example already. There's a reference in the Roud Folk Song Index to a traditional set recorded by Edith Fowke from Owen McBride of Ontario, which begins, One morning as Jack walked the beach up and down, but I don't have any further details. Meanwhile, here are some additional references:

The DT set, THE FIRESHIP, credits no source of any kind, but has the same first line as the example in Silverman's The Dirty Song Book (1982), and may derive from that book, or perhaps from a commercial recording (of which there have been several; a couple of American "songwriters" even copyrighted it at some point). More information, together with an early version of the story from c.1620, Watton Town's End, is given in this earlier discussion: Fireship/One of the Roving Kind

Also in the DT: THE ROVING KIND  Modernised text as recorded by Guy Mitchell.

There is also some discussion in this thread: She Had A Dark & A Rovin Eye

There is an entry at the Traditional Ballad Index:

Fire Ship, The

There are a couple of broadside editions at Bodleian Library Broadside Ballads; here is one:

Covent garden ramble Printed between 1819 and 1844 by J. Pitts, 6 Great St, Andrew street 7 Dials [London].