What stitch are you using around the applique? I use soft fuse to fuse the pieces to the base fabric and a small, close machine blanket stitch. I have done a lot of quilt applique and no tunneling. I use cotton fabric and cotton thread. I have never used or needed a stabilizer. I am sorry you are having trouble. I wish I could look at your applique for a diagnosis. Perhaps you could take it to your local quilt shop or quilt guild. Please don't give up on machine applique.