What kind of interfacing do you use? I want to do some applique on a quilt I'm working on and this sounds like an intersting way to do it. I'm directing this question to CoriAmd

you are not catching enough of the fabric to get past the fraying-
widen your satin stitch- or fold under the edges (either by adding interfacing (right sides together sew all the way around- slit the interfacing- turn right side out- place applique- then satin stitch around the finished edge)
or if you could try some fray check along the edges- and widen your satin stitch so it grabs more of the appliques

This thread has been really helpful, thanks for all the great tips, keep them coming!