The Molly Bawn I have heard doesn't have anything to do with being mistaken for a swan and shot by one's lover. I have heard that song in an English version sung by Chris Foster but it didn't mention any Molly. I think the lady in question in that song is named Polly, and she seems to survive the ordeal.

The Molly Bawn I know or have heard begins:

Tis I am sad and lonely
All in the distant west
Happy dreams of boyhood years
Last night disturbed my rest

Then there is something about boating on a lake with Molly Bawn. The tune is completely different from the one about the short-sighted hunter.