I don't know anyone who doesn't rinse their dishes before putting them is the soapy water to soak until they are washed. You mean you just put your dishes in the water with food still on them????? How disgusting....
I think you need to move the drainer to the counter top and keep the side that has a disposal, if you have one, open for food debris to be rinsed off before washing. Then while the dishes are sitting in the hot soapy water you scour out the other sink so it is sanitary for the clean dishes to go in for rinsing. You turn the water off while washing and then when the rinse sink is full you turn the water on and rinse all those dishes off. Putting the rinsed dishes onto the drainer.
Another way of doing it is like child care. You get everything ready but rinsing the dishes off first, put the dishes in the hot soapy water, scour out the extra sink, rinse it very well, use the stopper and fill it with medium temp or cool water. Add 1 Tablespoon of bleach to the rinse water to sanitize the dishes. Then as you wash you put the clean dishes in the rinse water. It helps if you don't use a ton of dishsoap in the wash water. The rinsed dishes are sanitized and if you let them air dry they are the cleanest they will ever be.
If you are drying your dishes you are not allowing them to stay clean. It is very unsanitary to dry dishes. It is not allowed in child care. The rag gets your skin oils, your dead skin cells, allergens from the air, etc...they are really not very clean.
I would also like to say, doing dishes any differently than how you are currently doing them is not going to make any difference in your water bill.
If your water bill is so exceedingly high I would call the company and ask them what the rates are for water. You may have to pay a base rate not matter how much you use. One of my friends can fill a pool every day, take multiple showers every day, she pays $30 per month for all her water, if she drinks one glass of water per day she still pays $30 per month. We pay a basic rate and then a usage amount for anything over XX units of water. So if we use excess we get charged for it.
SO when calling the company ask them if they can tell you the basic average usage on your current address for the past couple of years. Then you can see if your usage is less, more, the same, excessive, etc...than all the other people that have lived there in the past while.