I was born and raised in Fall River, and I would say that everyone in town knows the song, but I never met anyone who knew any part except the chorus. My father taught me a different version of the chorus, which is probably not in the original, to say the least. It makes me wonder if there other extemporized verses out there:

On the Old Fall River Line (2x)
A bum came in my backyard to see what he could find.
He found a pair of shorts, size 49.
Said, I bet my life
They will fit my wife
On the old Fall River Line.

I know, not a high point of Fall River haute culture. By the way, I recently learned that my grandfather at one time worked on the Old Fall River Line.