Aside from increasing stitch length and decreasing pressure on the foot (if your machine has this adjustment), it can help a lot to heavily starch the fabrics before layering. I use a 1:1 solution of Sta-Flo liquid starch and water. Starching stabilizes fabric so it is much less likely to stretch or distort while you are sewing. Heavy starching helps especially if you are quilting on the bias of the fabric (which it looks to me as if you are doing).
Also, be careful not to stretch the fabric as you quilt. If you do, when you go to cross the lines you will have a pucker right before the line. This is because the fabric as stretched as you sewed. Again, heavy starching helps prevent this as the fabric is much less likely to stretch if it is starched.
Starch should not interfere with your machine, and it washes out.