Dear nerotensia,
First of all, your sketch is terrific. And, your problem is that you are folding over a seam and stitching it down. It's stretching out of shape.
Now if you want to do this, there is a new tape that can be ironed on (carefully, avoiding stretch), then sew as you have been. I should tell you that I would never allow my students to do this, because of the very problem you have had.
But for a perfect finish, cut a piece of self fabric on the bias, or crosswise for knits, and sew it to the edge, holding (but not stretching) taut the binding, not the garment. Understitch the binding, then sew the second side to keep it in place. If you examine good RTW, you'll see that that's how it is done in industry, even though it looks like it was just folded over.
Teri