Hi Tony,
The answer to both your questions is yes, if you leave the tank full when towing you are adding unnecessary weight to the caravan, it only take a couple of minutes to open the drain valve, also when you prepare the van for winter storage you should drain the tank as well, if you have water left in it and you have a freezing cold night there is a risk of the water in the tank freezing and splitting the tank open resulting in a very expensive replacment tank or no hot water.
Hope this helps
Graham (Ponty)