The "nicked my jaw twice" part makes me wonder if you're rolling the blade at the end of the stroke instead of keeping the spine close to your face the whole way. On the left side I switched to holding the razor with my thumb on the tang and my first two fingers sort of pinching the shank, almost like a pencil, so I can pivot my wrist without dragging the edge sideways. That grip lets me shave up and across instead of straight down, which is the move that always got me near the jawline. Are you stretching the skin with your free hand or just relying on the blade to flatten it? Because on the left side I have to pull my jaw up and toward my right ear or the skin bunch and the razor catches. Also, how many passes are you doing before you feel the pull? Two nicks in one shave sounds like the lather dried out on that side.

My granddad ran a cardboard box setup, swore by it.