I would not assume 'the indicative pronoun "I" in the objective' is for the sake of rhyme; many if not most people do not know the correct use of "I" and "me", and furthermore, many if not most don't really care. And then there are certain dialects in which "I" is generally used rather than "me" in the objective (e.g., "Oh, never mind, I'll go and try,/Perhaps she might but fancy I").

I would think the bulk of traditional folk lyric would be painful for anyone who's disturbed by non-standard or just plain bad grammar.