I am a Bob Dylan super fan, but my go-to version of Don’t Think Twice It’s Alright is by John Martyn.

I love In the Morning by David Kitt. The song was originally by the Bee Gees.

I’ve been listening to Green River by Eilen Jewell a lot recently. I prefer it to the Creedence original.

Tainted Love by Soft Cell is a cover. The original was sung by Gloria Jones and is pretty forgettable.

P.S. How can a cover not be of an original song? Are you just trying to exclude versions of folk songs?