A friend sent me this picture yesterday.... Yuck! I told him/her/it to not skin it, that it quite likely has rabies as outlined in government propaganda. He/she/it took my advice.
Likely it is a very good idea for all trappers to get their rabies shots preemptively, I got mine years ago and was told I was good for life, but at the first sign of trouble they want to immediately pump you full with more crap rather than assess you to ascertain that you are in the safe zone... but it is for your personal benefit.
So, according to government literature, if you get a coyote or fox in such condition, do not skin it. At the very least burn the thing, the government would like you to report it and claim they want you to turn the animal in, however, the last time I wanted to turn a definite diseased animal in, nobody wanted it and we were stuck with it in the car until I got home from the hospital whereby I burned it immediately.
The End
(Wasn't that a nice story?)