I always understood a 'tarrier' in songs to mean a lazy individual, one who wouldn't work. To 'tarry' as in

Let him go, let him tarry
Let him sink or let him swim.
He doesn't care for me,
Nor I don't care for him.
He can go and find another
and I hope he will enjoy,
For I'm going to marry a far nicer boy.

means to wait. I'm almost certain about this definition, so it's up to youse.
B.