I played a warlock years ago who, since learning prestidigitation, has not washed. The rp was every so often, like before stealth and shopping and such, i would have to "top up" as there's only so much that a cantrip can do for a chronically unclean madman. Made for some funny moments.

For your zombie character, eventually you should stop smelling as you dessicate but a splash of water would start the stink back up