My parents sang this a lot, to the first melody posted by Bob. I ran it through the Concertina program but I hear the chorus somewhat different. The verses were sung in a slow waltz time but the chorus changed to something akin to 8/4 tempo. Imagine that the first three words 'Why do I' are the 6,7 and eighth beat of the intro, then 'weep' would be the first beat then count '2,3,4,5' Why do I   'Sigh'(2,3,4,5) My love's a- Sleep (2,3,4,5) so far a Way (2,3,4,5)
In other words, 'weep', 'sigh', 'sleep' and 'way' are the first beats in each bar and very measured. Hope this makes sense ..I can still hear them singing together.