Another vote in favour of keeping Auld Lang Syne, even if it is the version that most people at least know the tune to, if not the words. (Recent story in "The Scotsman" newspaper suggested that even among Scots, only 3% of people know all of the words of the 2 verses usually sung - or when to cross hands and the like!
I love the older tune, and know all 5 verses, though I wouldn't necessarily venture to sing them all at the end of a raucous night. Among friends of a similar leaning, different story.

But if you are seriously looking for another song to finish the night then Alan Bell's "When First We Met" is a good one (NOT Irish!!!!! as I have heard it introduced!!!!!!) or "In Friendship's (Freenship's) name" if you can again get your tongue around the Scots words, or Ian Sinclair's "Tak a Dram Afore ye go" (but don;t take that literally if you're driving!) And various Irish "Parting Glass" songs (but same provisos re who is the designated driver!)