Okay, so option number one, and the one that looks easiest, is to try to make the water keep going through your last loads. After all, who wants to empty the sink, wipe it down, and refill it with water. Failing to empty cold and dirty water has several undesirable consequences you need to be aware of.
1. Grease and food left in the water leaves residual dirt and germs on your dishes.
2. Cold water is inefficient for removing stuck on food and grease from your dishes.
3. Leaving the wash water tepid and dirty doesn't actually save you from filling a new sink of water. You will more than make up for that water loss by trying to rinse off grease and germs during your rinsing.
Taking the few moments to refill your sink when the water becomes cloudy and cool may seem like an inconvenience, but it truly can save you time and more importantly preserve your family's health.