Efficient laundry is not any type of hanging fruit regarding climate change. It has nothing to do with climate change. Doing laundry some other way has zero impact on the future climate. Powerful natural forces will continue to change the climate, as they always have and always will.
But it will have a large effect on how we live our lives. My chore as a boy was hanging the wet clothes on the line, and collecting them. They smelled great. But a great many of us don’t live where that space is available. And try it in the winter or the rainy season. My high tech machine washes a load in much less water than back then, and most places do not have serious deficiencies of water. Phoenix might benefit from shared machines and clotheslines, but Minneapolis won’t. Especially in winter.