I found a copy of LET ME IN THIS AE NIGHT in Ancient and Modern Scottish Songs, Heroic Ballads, etc., Second Edition*, by George Paton (Edinburgh: James Dickson and Charles Elliot, 1776), page 167.

--Despite that The Traditional Ballad Index has marked it "Earliest Date: 1976" !

The words are nearly the same as those given in the DT, so I won't repeat them here.

There is no musical notation in that book.

*I haven't seen the first edition.