You'll know that it's already been extensively discussed here; this new thread will probably just attract repetition of what has already been said. I can add a little, though.

The earliest example I know of is in Sam Henry's Songs of the People, p 269, from Bridget Kealey, Dungiven, 1936. Frank Harte told Ewan MacColl "The Galway Shawl is not a very serious song and has always been sung at a kind of popular level and of late was adopted by the showband crowd." (MacColl and Seeger, Till Doomsday in the Afternoon, Manchester University Press, 1986, 218-220; note accompanying a set recorded from Shelia MacGregor [Sheila Stewart]).

I don't find any reference to broadside examples, though it's not unlikely that it was issued on songsheets at some point. Clearly related, though, is a song called The Red Plaid Shawl, which has a very different tone. It reads like a parody of the sentimental song. Copies can be seen at Bodleian Library Broadside Ballads. None are from identified printers, so are undated; but the song appears in catalogues issued by Sanderson (Edinburgh) and Such (London) so would likely be of the second half of the 19th century. Galway Shawl, if it was indeed the model, would be a little earlier, but probably not by very much.

The red plaid shawl