I usually cut mine from the back so I'm only cutting the backing. Then I brush the loose bits out of the fur from cutting initially. Then I trim the fur itself from the seam allowance area and brush the loose bits out of the fur again. The more loose bits you can brush out, the less that gets into your machine. Use a good heavy needle in your machine when you sew and take your time, do not push the fabric through.
Fur is fun stuff to play with. Keep your design simple. The fewer seams the better. Can't wait to see what you're making.