When I make anything for the chins I sew it inside out and then leave just enough of a hole to flip it rightside in and stitch just that little area. This way 98% of the item has no exposed stitching
When I hand stitched the cuddle buddy, I just used the back stitch. I sewed it in a similar fashion to how sinfulcindylou described. There was only a small area where the thread was exposed.