Helen,
Try using twill tape along the seam allowance on the wrong side of your quilt top before you quilt it. I buy it at Joanns in the hem tape/blanket binding display. The color doesn't much matter because it will fall within the seam allowance. Twill tape has no stretch or give and will stabilize those cross-grain sides for a wall hanging panel.
I apply it to the wrong side as below.
Jan in VA