Glue the binding on before sewing. Press with a dry iron. Then use the clover clips or pin the binding to prevent shifting. This includes the mitered corner. If the corners are too bulky, snip a small triangle from the quilt corners. Snip a little at a time and only snip enough to make the mitered corner. There really shouldn't be any bulk at all to the corners. Try using the serpentine stitch with a shorter stitch. Also, I prefer sewing the binding onto the back first then bringing the binding to the front when I use the serpentine stitch. It allows me to control the way the front binding will look. Practice on a sample to see where the left edge of the stitch will fall with the binding. I sew the left edge directly onto the edge of the binding, then it is uniform all the way around. I use an 80/12 needle and size 40 thread. I don't use the walking foot with the serpentine stitch.
Once the quilt is washed, you may not notice the wavy edge at all.