Run it through the drum carder. Why this hadn't come to me sooner is beyond me but I ran a bunch through and am thrilled with the results.
I can actually spin it now. Does it still shed? Heavens yes. Is my lap littered? Not as much. Running it through seems to have knocked quite a bit of dirt free and while I still get covered it isn't as bad. My fingers don't turn black nearly as quick either. That's always a plus. I ended up doing 8 bats and have barely made a dent in the box. I will be spinning this for a long time.