We have warm water coming out of our cold water taps for about 10 seconds. Happens in all the taps. Sometimes run out of hot water after 1 or two loads of laundry. We've had the hot water tank checked and they say the tank is fine, but it's a splitter on one of our single handle faucets. All but one of our faucets is single handle. We don't want to replace them all, so how do we tell which one may be the faulty one?

It might not be a crossconnection in your faucet. Put your hand on the cold water pipe feeding your water heater and tell me how far the warmth runs on the pipe. This is called convection and you could be pulling this back into your cold water system.
Let me know, Tom