I think some people tend to unconsciously correct for their own style as they go, and remember the songs the way they feel comfortable singing them, so it's not always an affectation when the words seem different. It can jar, though, if you have always sung it in its "incorrect" grammar.
But yes, written grammar and spoken grammar are often different beasts. OTOH "isn't" is perfectly correct written or spoken, so wouldn't need to become "is not", and "can't" similarly wouldn't need to be "cannot".
The real difficulties for me come with songs intended for another accent (Scottish, any of the American accents, Geordie etc) and I never manage to cope with "ain't" as it's not a word I usually use.
What I really, really hate is when a (non-trad) songwriter forces grammar into a non-conversational OR written form to make a rhyme. No examples spring instantly to mind, but there have been a few!

Anne