Windshield wiper motion should help you a lot. I was getting better shaves with less irritation once I started using something similar. The way I've done it gives the blade more of a scythe-like motion. It's a simple wrist flex, easy to learn & do consistently. All blades cut better when they have a bit of an angle or sideways slicing motion. Just be careful you don't take it too far or your cheek might land in the sink :)
Edit: I just realized your main razor was a square tip. May not be all that easy after all. Mine are French tip with the corner slightly rounded so I don't have to worry about it digging in. I doubt I'd have the nerve to try it with a square tip, although using it on an angle shouldn't be much of an issue.