We normally travel with our fresh water tank completely empty. Contrary to what other say, when I have water in the tank, I can feel the difference pulling the trailer. There have been occasions when I've forgotten to drain, or we had water in the tank at home and I didn't drain when we left for a trip. One mile down the road and I'm always saying to my wife, something is wrong, it just doesn't feel right. She says every time, "Did you drain the water?" Duh. I can tell the difference. We carry a 6 gallon jug in the camper with filtered water from home. We use this for my wife's coffee and to use to flush the toilet if we are on the road.
Now about the black and grey tank. We travel with the grey tank as empty as possible. But if we end up using the sink, we have no problems letting water go down the drain (there again, using that 6 gallon jug.)
The black tank is a little different. When flushing the tank, it's always back-flushed with the grey first, then empties completely. Once we know it's cleaned out as humanly as possible, we'll add 2 or 3 gallons of water. If we want to clean the tank a little, we'll add a bit of Dawn Dishwashing Liquid, and sometimes some bleach. We only add bleach if we know we can refill the tank and dump it all. I don't like bleach water in my tank too long. I don't want to risk the valve seals drying out, or crumbling because of bleach exposure. Bleach is caustic if it sits long enough. So, when using bleach, I have to be able to rinse the tank and flush within a short while. We'll do this sometimes when traveling and moving from one camp ground to another, or do it at home when leaving, knowing that afternoon we'll be in a full hook up camp site and I can flush the tank then.
If using Dawn or Tide or something like that in the tank, all you need is a couple gallons of water and a nice generous amount of Dishwashing or Laundry soap. All the sloshing around in the tank when driving will pulverize the inside of the tank and clean it real nice. No need to dump it, just use it and dump when full at the campground.
We have, on occasion, traveled with a few gallons of water in the fresh tank. But we avoid it as much as possible.