I would recommend a lightweight thread and a smaller sharp needle in addition to the stabilizer, assuming you are not working on a knit. Also, do you loosen the tension so the top thread wraps to the back side of the fabric? I do know some fabrics are just harder to get even stitching on (denim and other loosely woven gpfabrics in particular!), but you should be able to get a very good satin stitch on your machine.
Pam