You can't turn zombies off except for by setting the single lunar phase to anything but full moon, or, to use a mod. Turning off zombies is a simple tuning file.
There are a few things that change around fullmoon, mostly werewolves. Fairies get a better moodlet, supernatural fans and a few other traits get a lunacy moodlet which makes them stop everything they are doing and slap people or hug them every now and then. There are a few more wishes related to full moon that can really add up - go out during a full moon, catch a fish during a full moon, harvest a plant during a full moon, as well as visit the grave yard. I know there are also other collecting wishes related to full moon, though I haven't expanded my current character very far. I've been playing on full moon single lunar phase for the last few days, and the zombies did get rather annoying.