Oh, and by the by, I've always sung "And the days of auld..."
Guess maybe it is a US tradition.

I think the reason we get "Old Lang Zyne" is because of the proximity of the 'g' ending 'Lang' and the 's' beginning 'syne'. I think we US singers tend to be less precise with our consonants than our UK cousins, and running them together produces the 'z' sound heard here.

Blessings,
Barbara