Here is something I strongly recommend for you to use when you wash dishes with. It's Biwako Fukin. It's a washing cloth a couple make manually. You don't really need detergent even your dishes are oily. The cloth absorbs oil and somehow, you can rinse off the cloth quite well with water.
Many shops selling the cloths for the proper price sell them out quickly, but if you try to be patient, you can order them for the proper price. Otherwise, you would have to pay triple the price.