My experience is that hand dishwashing done carefully with a plugged sink and in bulk, not one dish at a time, uses less water and therefore less electricity than a dishwasher. If your water is solar-heated, hand washing is even better. But modern dishwashers are more efficient than hand washing with the water running constantly.

I’d guess that LG is about as good as you’ll get. I recently bought a LG washing machine and its electricity use seems as economical as it can be. It has a steam cycle, which is a real energy saver for lightly soiled clothes.