Here is the Lanarkshire ("Carnwath Mill") version that I happened upon while looking for the more usual lyrics a few years back: can't remember now which website I got it from.
And yes, we always play the tune as a final item, after Auld Lang Syne, in one of the ceiidh bands I'm in.

We're No' Awa' Tae Bide Awa'
(Lanarkshire variation)

Chorus:
For we're no' awa' tae bide awa',
   For we're no' awa tae le'e ye,
For we're no' awa' tae bide awa',
   We'll aye come back an' see ye.

As I gaed doon by Wilsontoon
   I met auld Johnnie Scobie,
Says I to him will ye hae a hauf,
   Says he, "Man! That's my hobby."
Chorus:

So we had a hauf an' anither hauf,
   And then we had anither,
When he got fou' he shouted
   "Hoo! It's Carnwath Mill for ever."
Chorus:

We wandered doon the street again
   We cleekit unco cheery,
When John got hame his wife cried shame,
   I see you're enjoyin' your hobby.
Chorus:

Of a' the friens that ere I kenned,
   There's nane like Johnnie Scobie,
His hert is leal, he's true as steel,
   An' a hauf is aye his hobby.
Chorus:

So whenever friendly friens may meet,
   Wherever Scots foregather,
We'll raise our gless, we'll shout
    Hurroo, It's Carnwath Mill for ever.
Chorus: