I addressed this subject in my post How to Reduce the Cost of Water Heating, Water Softening, Water, and Sewage. You can reduce washing costs by using cold water. It is more efficient to do full loads of laundry. You may be able to use a shorter wash cycle. You may be able to use less soap than what you are using. If you are really ambitious, you can make your own laundry soap. You can at least look for sales on detergent or buy house brands.
A gas dryer is cheaper to operate than an electric dryer. It isn’t cost beneficial for us to get rid of our existing dryer and put in gas one, particularly since we don’t plan to be in our house for more than a year.
The really ambitious don’t use a clothes dryer. We don’t fall in that category. We did have an outside clothes line, but found we had more allergy problems when we used it. We find it inconvenient to hang up our clothes inside to dry. Our dryer does have a moisture sensor which prevents us from overusing the dryer. You can use your dryer for a shorter period of time if you shake out clothes before you put them in or take them out to hang before they are completely dry.