Satin shading is very similar to old school hand embroidery where you interweave long stitches along the surface. Most traditional fur is done in narrow patches of satin-style stitches with feathered sides.
When you use this method, you are likely to end up with quite a lot of color changes, but the end result is well worth putting in the time required to stitch out your design.
You need to switch off all underlays and make sure you set your start and end points correctly to avoid lines behind your objects.