It sounds like you are having bias edge issues. Some solutions are starch starch starch until the piece is quite stiff....(i make my own starch with 1/2 liquid starch and 1/2 water in a spritzer bottle)......use washable glue dabs to "baste" the pieces together AND overly pin the parts to keep them from moving.....use a walking foot to help you not stretch the fabric.
Maybe this is not the problem....but that was my first thought RE: the problem.
GOOD LUCK !! it's a beautiful pattern so i hope you find the right solution.