The traditional way of adjusting a first version song in this way has been some introductory verse of the form

As I was a walking one morning in May
To hear the birds whistle and see the lambkins play
I espied a young damsel, so sweetly sang she...

I suppose this is one way that songs get rewritten - but I doubt if its that common among traditional singers to have this kind of feeling you can't sing in the person of someone of the opposite sex.

I suspect failing memory is the engine that drives the folk process. Singers make up new words to fill in the gaps; and other singers rewrite verses where the replacement words don't seem to make too much sense.

Another source of change is the fact that some combinations of sounds make us stumble when we meet them, so instinctively we adjust so as to avoid them. That kind of thing is especially likely to happen where people have some kind of speech impediment - for example, a lisp or a stammer. But in some measure it applies to everyone. As a song goes around the rough edges get rubbed off.