Starting with the water, if you have your manual check to see the winterizing instructions it gives. You may be lucky and the previous owner opened the drain valves to prevent freeze. Don't forget the drain valve on the water heater as well. Find them, close them and try filling again. You should be able to see the water draining if the valves are going, if the leak is difficult to find you may have a loose fitting or burst pipe and will have to backtrack it to find and ix. If you don't have a manual search this site and you'll see threads on how to find one. For the electrical, my lights run off of the battery even when plugged into power. My electrical outlets however only operate when plugged into 120. Try plugging the trailer into power and test your outlets. Welcome to vintage airstreaming....
"would you rather have a mansion full of money or a trailer full of love?"