Mac, from the worldburnsclub.com kindly sent me a message regarding Burns and Robin Adair.
"I believe the song you are looking for is "Phillis the Fair" - Burns wrote this piece about one Phillis McMurdo, the daughter of John McMurdo of Drumlanrig. Burns set the piece to the tune Robin Adair, referred to by him as "a crinkum-crankum" Gaelic tune. He obtained the tune from a "musical highlander in Breadalbane's Fencibles" - based in Dumfries in 1793. (Ref Complete Works of Burns - Dr. J.A. Mackay)