I would fill up your water tank at the dealer, use everything at the dealer (all taps, shower & water heater) then when you go to the campground use the hook-ups. This is probably closer to what you will do most days and will allow you to test the system both ways.
I would also do a dump (have water in both tanks) the day you pick-up (run water in all the drains to make sure they work and don't leak) to make sure that you know how and that it works and so you can check for any leaks and that you have a dump hose.
My system for traveling has been to fill up water tank at home, travel using the on board system, make camp, use what I need to then the next morning dump tanks and fill up water before hitting the road. I never know if the next day will bring me to a campground that has water or a dump station so I like to stay ready either way. If I have full hook-ups then I top off the water tank at night.