What kind of pattern is it? You said you pinned and stitched the border. Did you also pin the center part? Can you run down the length, doing a stitch in the ditch type thing, maybe on a sashing seam? If it is OK on the back, no pleats or bunching, do the opposite side of the quilt. Then across, etc. to stabilize the squares. Then I think you would be OK to quilt each block without worrying too much. If not, maybe you can do the charm tack method, where you pick a shape, like a big heart, little heart, and stitch here and there instead of tying.
Do the best you can, chalk the rest up to a learning experience, and quilt on!