Hello, Mysha. I don't know where your lines came from, but they seem to start in the middle of a verse. Here are the same lines from the DT, typed to show the internal rhyme (or near rhyme):

At Orenmore in the County Galway,
One pleasant evening in the month of May,
I spied a damsel,
she was young and handsome
Her beauty fairly took my breath away.

Cho: She wore no jewels, nor costly diamonds,
No paint or powder, no, none at all.
But she wore a bonnet
with a ribbon on it
And round her shoulder was a Galway Shawl.
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The overall rhyme scheme sure changes a lot in the course of the song.

Thanks for the explanation about the meaning of a Galway shawl.