Actually this is a great question, since most animal grooming tends to involve a mouth.
How much an oder they put off I guess is questionable- Ax certainly doesn't think they stink, and I don't know how much they sweat.
Also does morphing help: I know it's exhausting, but if you run 5 miles, morph and demorph, are you less grody? Don't know
I would amuse an occasional dip in the river, but really beyond that, it's a puzzler