Burns seems to me to have been quite liberal in supplying copies of his songs and poems to others. I didn't copy down the name of the society it was dedicated to, but NLS MS 6302 (which I glanced at on a microfilm copy) is a big collection that Burns sent them. Burns scholars don't seem to know about MS copies of his pieces in the Douce collection at the Bodleian Library at Oxford. Captain Grosse got them from Burns and carried them to London and gave them to Francis Douce.
I've also seen it said that Burns took little interest in folk songs and ballads, which is complete nonsense as far as I'm concerned. We got the first full version of "Tam Lin" from him, as well as "The was a pretty may" (a version of "Dabbling in the Dew" that's on my website) and "A Waukrife Minnie" (17 come Sunday), SMM #288, that he collected from a Martha Crosbie.
I see no reason to believe he lied when he said "Auld Lang Syne" was the original song, and he got it from an old man's singing.