I was going to suggest using either spray starch, stay-stitching, tear-away or wash-away stabilizer (the kind you iron-on first), or have your tried turning your project upside down (put the more stable fabric down on the feed dogs, or lower the feed dogs so they won't engage the fabric at all. Or adjust the presser foot tension, or use a free-motion foot (
Free-motion Darning Quilting Sewing Machine... )