I have done this with guitar... but it may be easier for instruments than vocals (since vocals can vary so much more widely).
But, I would do this:
Leave the original verse in tact, no changes to it whatsoever. Then, arm another track DIRECTLY below it for recording. Then, play/sing some of the part before where you screwed up and some of it after you screwed up. Then, play around with fading/volume so that they blend properly. Then, downmix them into one file
Exactly. When it comes time to blend the parts of each track that you want to mixdown, you're going to have to zoom in on the track so that you can bring the level of one track up at exactly the same time that you're taking the level of the other track down. That way you avoid jumps or breaks in the volume of the final track.