I see that the French Collection also has Mrs Boswell of Auchinleck's Reel (which makes a nice air too, though a rhythmic one rather than slow) so this tune at least refers to the right family and is around the relevant period. But neither of the titles in French's anthology actually claims to be BY Boswell, whereas the piece in Gow ("Shelah O'Neal" in 6/8 time) does. But then, we know that Nathaniel Gow played fast & loose with the truth on more than one occasion.

So maybe these are written for the Boswells instead? Still interested to know more, if more is to be had.