There's a song with the same air "Sweet Cootehill Town". Maybe it was that? I think Sam Henry's book said the song pre-dated the Parting Glass.

SWEET COOTEHILL TOWN
As recorded by Reg Keating on "Songs of Cavan" (2015)

Oh fare you well, sweet Cootehill town, the place where I was born and bred.
Through shady groves and flowery hills my youthful fancy did serenade.
But now I’m bound for Amerikay, a country I never saw.
These pleasant scenes, I will always mind when I am roving far away.

The pleasant hills near Cootehill town where I have spent my youthful days,
Both day and night, I took delight in dancing and in harmless plays;
But while I rove from town to town fond memories in my mind shall stay.
Of those pleasant happy youthful hours that now are spent and passed away.

Perchance kind fate will reinstate and fortune’s face upon me smile
To safe conduct me home again to my own dear native Irish isle,
When all my comrades and friends likewise will gather round and to me will say:
"We’ll sing and dance as in days gone by; you are welcome home from Amerikay."