What fusible are you using?
It looks to me as if the fabric is fraying. Are you catching enough of the edge inside the satin stitch? You want as much of the stitching as possible to be on the applique, as little as possible on the background.
I think the problem may be that you are catching only a thread or two of the applique within your satin stitch. When it's only one thread and it's on the straight-of-grain, looks like the stitching is pulling away from the fabric (because that one thread gets pulled away from the applique).