Did you stabilize underneath the minky before appliqueing? If not, try stabilizing. A good quality tear-away or cut-away stabilizer would probably help. JoAnn Fabrics carries that kind of thing, and probably also Walmart. It could also help to add a single layer of tissue paper to the top, between the applique and the minky. You would have to carefully tear the tissue paper away afterwards, and might need tweezers to get little pieces out. The small amount left inside the applique will just disintegrate in the wash.
It would help to see a photo of the problem.
One other thing. Do you mean you can't get the stitches close enough to each other? A solution to this is to sew over the edge twice. If you can get a longer zigzag to hold the pieces down, then you can just go over it again with the slightly shorter zigzag as many times as necessary to get the build-up of thread you want. I can't tell from your description whether you are using a long zigzag or a short zigzag (which would be a satin stitch), and whether by "close" you mean you can't get close enough to the edge of the applique to catch it in the stitching, or that you mean you can't get the zigzag stitches close enough to each other to create a satin stitch.
Last edited by Prism99; 01-27-2017 at 12:06 PM.