There are several versions of a tune called Within a Mile from Dublin. I don't know if it's the same one you are looking for.
Also, I don't know if you are into ABC notation but there are scads of transcribed Sharon Shannon tunes (without titles, unfortunately) in the Adrian Scahill collection, his file Manuscript 2.
Connaught Man's Rambles I only have as a jig.
I don't believe we can post GIF files but if you wish to e-mail me at: radriano@consrv.ca.gov I can get the music to you. I can only receive e-mail during the week so I probably wouldn't get anything to you until next week.