The Roud Folk Song Index places this one on its own at number 2384, while 'Polly on the Shore' is at number 811. I don't think there's any connection between Joyce's fragment and 'Polly'.

The 'Polly' text in the DT (beside the ludicrous misattribution, which has been pointed out several times over the years but has still not been corrected) is a bit of a dog's breakfast cobbled together from a couple of mis-heard revival recordings so far as I can tell. The lines in the first verse

... A privateer to trade
To the East Indies we were bound to plunder the raging main
And it's many the brave and a galliant ship
We sent to a watery grave

don't appear to belong to any known traditional form of 'Polly', and were probably made up by Trevor Lucas back in the 1970s. That takes the East Indies out of the equation, and the remaining similarities are only generic. Joyce's text probably appears on broadsides, but since it appears to be from the middle of something it may prove difficult to track it down on purpose. It's the sort of thing that Steve Gardham might recognise, though.