In pixel art, you don't really have the space to make good-looking individual strands of hair. Try working in clumps instead.
On the other hand, an overabundance of texture was a staple of that particular book's art style, so maybe you're on the right track, if you're out to imitate it! In that case, however, I recommend taking the time to block in the shadows before you do the fur texture, so that when you texture, you can incorporate the shadows.