Donald A. Low (University of Sterling)in "The Songs of Robert Burns" (1993) agreed -- this is Burns. Don't know why the song doesn't appear on the Burns Country database. On page 520, Low wrote:

"Burn's revision of a song in David Herd's Ancient and Modern Scottish Songs (1776; reprint 1869, vol. 2, pp. 182-3). Burns has made a number of minor changes, and for delicacy's sake has substituted his own lines 33-6 for
    A pair of white legs, and a good cogg-wame,
    An ye'll come hame, &c.
in the original.
   Stenhouse states that Burns communicated the air printed in the Museum, but that the song was formerly sung to Bab at the Bowster (Illustrations of the Lyric Poetry and Music of Scotland, p. 340)."

Low's book has texts, melody line notation, and much anecdotal info -- a *must* for any Burns aficionado. Out of print, but available on Kindle for $23. http://www.amazon.com/The-Songs-Robert-Burns-ebook/dp/B000SIZET6