Spray starching now would likely help a lot. Lay the quilt out on a large sheet and spray from the outside towards the center to minimize overspray. Let the spray starch dry, then spray again. Do about three layers of spray on both top and backing.
The problem is that the fabric is stretching as you sew. You get the pucker at the previous sewing line because there is no more room for "give" at that intersection. If you can stabilize the fabric so it doesn't stretch while you sew, that will help a lot.
Also, do you have the ability on your machine to decrease the pressure on the presser foot? If so, easing up on that pressure may help the sandwich move so you don't have to push it through.
Have you lengthened your stitch? Are you using a walking foot?