Thanks for the suggestions. Tonight I plan to first try a larger needle, change the fabric and pay attention to my speed. I really hope one of those fixes my problem! I'm convinced that needle flex is the root of the problem.
I don't think it's the speed because it starts to skip stitches immediately when I push away from me, and it will fail to make any stitches until I stop moving in that direction. Sewing at any angle away from me gives unpredictable stitches, with straight away being the absolute worst!