I do a whole lot of raw edge fusible applique. I have always used Wonder Under and I never cut out the background behind the applique. I use a straight stitch and invisible thread and stitch about a 16th of an inch from the edge. When you use invisble thread the spool needs to be upright and the thread needs to come off the side of the spool.
The other technique I use is black blanket stitch. My machine has a nice blanket stitch and it really gives your applique a nice folk art look. The Wonder Under is enough there is no need for an additional stabilizer.