There's no real evidence that "A-Rovin'" itself is unusually old, even if the general idea appeared in a completely different song in 1607.

The earliest polite version seems to be in [William Allen Hayes, ed.] Selected Songs Sung at Harvard College: From 1862 to 1866. (Cambridge: pvtly. ptd., June, 1866), pp.30-31.

So it could be a clean song that got bawdified and shantyized rather than the other way around. However, Whall gives the three tamest stanzas of the anatomical version as having been part of the shanty repertoire of the 1860s.

Harlow prints part of a version he presumably learned in 1876.