There is a song listed in Steve Roud's Folksong Index as The Carrick Courtship, which opens As I roved out one morning fair, it being on the first of July. No Roud number given. It was collected by Hugh Shields in 1968 from one Miss Madge McRory, of Glencolumbkille, Donegal. The recording is held by the Ulster Folk and Transport Museum, where its index number appears to be 6830. So an enquiry to them might put the matter at rest.

BTW., There's a Carrick in Donegal, which is only about 10 miles from Glencolumbkille, so I'd guess it's a local song from round that area.