The refrain doesn't fit but the stanzas do. There are several old songs that use the stanza tune.
Is the refrain an add-on or borrowed from another song?
Is the verse similarity coincidence?
The verse form also makes it easy to float verses from one to another.

I will go so far as to note similarities, but I'm not going to spend time studying relationships. In any case, I don't have access to enough material.