How are you fastening the applique pieces to your background? You need to use at least 2 pins (preferably applique pins, and preferably put in from the back) to secure each applique piece so it doesn't shift while you are sewing.
I agree that it sounds as if the thread is being pulled too tight as you sew.
Are you using freezer paper on the top of the applique piece, or underneath (ironed to the wrong side of the applique)?