There are plenty of songs about sailors from that period, and songs that sound as though they might have been put into the mouths of sailors, but no records of anything being sung by sailors aboard. For songs about sailors try the Roxburghe Collection or the Pepys Ballads.

The Hugill reference is completely discredited. It is based upon the theme of a song in Heywood's 'Rape of Lucrece' but the theme 'I placed my hand upon her .....' is common to many songs. Other than the theme it has nothing in common with 'A-Roving'. It certainly wasn't a shanty. 'A-Roving' is a much later shore song adapted to shanty form.