How do I fix crooked binding, or should I just start over?

Just redo the wonky sections. Ripping a third of the binding off a perfectly fine quilt would be punishing yourself for no reason, and since it's a straightforward machine-or-hand stitched cotton binding, the seam where the good section meets the new section will disappear once it's done. Pick a stopping point in the middle of a straight run rather than at a corner if you can. Unpick the front stitches, then free the back, then pull the old binding off and re-fold your raw edge before joining the new piece to the old with a diagonal seam, same as when you first joined the strips. The edge won't distort as long as you take the stitches out gently and don't drag the seam ripper sideways thru the fabric. If a stitch resists, back up and go from the other direction.

Been there with late night binding and it's never good. On the pucker question, before you unpick anything try misting the wonky spots lightly with water and pressing from the back with steam. Cotton bindings settle down surprisingly well and you might salvage parts of it. I keep a handful of little binder clips from the office supply aisle on my sewing table now, and I flip the quilt over every ten minutes or so when I'm doing the final pass. Cheap, nothing fancy, but that flipping habit is what saved my last quilt after I discovered half the back was unattached the morning after a very sleepy session.

Hand stitched binding rips out way faster than you'd think. I stopped using a seam ripper for it after a fix-up job on my nephew's quilt, now I just slide a fine needle under a stitch, pop it, and pull the thread. Every few inches the whole section just unzips and there's barely any stress on the edge. Also the cat angle tracks, mine likes to sit directly on whatever I'm binding and then acts offended when she gets pinned.

The needle trick is the way to go, agree with that. When I redid a bad stretch on a lap quilt last winter I popped the stitches by hand while watching tv and had the whole bad section off in under an hour, no snags in the fabric at all. One thing that helped me was marking my stop point with a pin on the back side so I knew exactly where the old binding was still sewn down. For the back side wandering, I started folding the binding over and checking every inch against the fold as I go now, then clip, then stitch. Slow but it works.