Sanitation is always a major problem after any type of major disaster that dislocates a large number of people. Having clean dishes can help keep you healthy. I have often used clean sand to scrub hard dried on food off my plates and cooking utensils. Sand will remove the hard dried stuff but it will not cut the grease. The best sand comes from lakes or streams beds.
A simple way to remove the grease is wood ash. Take clean wood ash from your fire pit and mix it with a small amount of hot water, creating a paste. Smear the paste on your greasy dishes, pots and pans. The lye in the wood ash will cut the grease. Be careful, the lye from the wood ash will make your hands dry if left on too long. Rinse the dishes in clean hot water.
Your ash needs to have come from a fire that burned only wood, no garbage. You need to avoid any fire pits that have burned plastic or other debris. The best idea is to start a new fire pit, so that you know that only wood has been burned in it. This is to avoid contaminants that may harm you. Hardwood makes the best ashes for doing dishes
All water for washing any utensils that will come in contact with food must be first treated to destroy disease causing organisms. The water should be treated the same as drinking water.