Dave Burland sings a version (don't I remember having heard that it was Nic Jones' version?) which has an extra (first) verse:

I'm going for a soldier, Jenny,
Going across the rolling sea.
I've got a shilling from the King
That they say has enlisted me.
I'm off to fight for the army,
As a Lancashire Fusilier,
Rolling my musket in my arms
Instead of my Jennie dear.

and it doesn't have a chorus, either. As to whether it's sung to Star of County Down I can't comment, for I don't know that tune.