It's not a matter of "bad" grammar; it's simply a matter of whether you speak/sing (or claim to speak/sing) in Standard English or not. I generally use the same grammar when I sing as when I speak. I therefore sing "She swum and she swum...." (Marrowbones) rather than the usual "She swam and she swam..." because I don't naturally use the word "swam". In the same way, I sing "And the blackbirds and thrushes sung from every green spray.." because it comes naturally to me to sing it that way.

Were I singing a song where, for example, the word "swam" was used to rhyme with, say, "ham", then I would make the effort to sing "swam" in order to preserve the rhyme.

As for "between you and I", that's not, to the best of my knowledge, a non-standard or dialectal form; it's just hypercorrection!