are you sure you didn't stretch your borders or cut them too long? did you measure through the middle or on the side? if your sure you did them right then you should do a straight stitch closer than 1/4" from the outside edge, that will stabilize it and hopefully make it behave so you can quilt, my other advice is to not use high loft batting unless you're on a frame so you can keep everything stretched tight and flat
good luck