Same story at my store, basically. Except my pf team is more cooperative within itself, sounds like. Tonight I am closing, which means zoning all of dry grocery and doing all the pf routines by myself. In addition I am sure they'll want some kind of project done. Keep communicating in a positive friendly way.
You said you don't have time to zone the coolers. Perfectly understandable. Neither do I! Communicate what you -did- do.
Also: you check dates on everything, every day? you may want to collaborate with your ctl on how much time you want to spend on that. For instance: if you are getting a truck shipment of food the next day, why are you spending all that time checking dates when that could happen during the push process the next morning? Everyone should rotate and cull as they're working anyway. One person can't check everything. Know what I mean?