You can try adding layers of spray starch to the backing to stabilize it more. Lay the quilt out on a large flat sheet, spray with starch, put a fan on to dry, then repeat several times.
Also, I am wondering about the needle. I have had wonky things happen when I have a bad needle. Recently I have found that my machine does better with a microtex (sharp point) needle. I put all of my universal needles away for now. Not sure if this has to do with the specific fabrics I am working with or what.... Anyway, what size and type of needle are you using? And is it a new needle?