The fur is worthless this time of year, you can't skin an unprime coyote and expect the fur (or lack thereof) to hold up.The flesh will turn blue/green and the fur will slip out in short order. Prime winter coyotes will take fur handling very well... so if you want a pelt to tan or even a tail to display make sure to kill you another one this fall.
The skull is the only realistic thing to keep right now- so take these guys advice and clean it up. Get that camera fixed too!