When we picked up our new TT last year, the fresh water tank was 1/2 or more full. Not sure if the dealers do that to test the system or if they ship them from the plant that way so there is water available to the driver during shipping.
The black and grey tanks should not have false readings unless it has been used a bit. "Slime" gets left behind on the sensors inside the tank (look like a small button) and messes up the readings and it's usually the black tank that causes it. Drivers are not allowed to use the black tank during delivery so the tank should have been pristine inside when picked up at the dealer.
As long as the black and grey tanks are reading okay after the tanks were flushed, you should be perfectly okay after that. False readings are the norm in RVs. You should always flush the black tank out as best as possible after dumping and before heading back home by using a built-in tank rinser (if so equiped), a Valterra clear backflush elbow or hand held flush wand. Some black tanks cause more false readings than other due to the shape and dimensions. Periodic cleaning using the Geo method or RV chemicals specifically for the purpose can help. The objective is to clean the sensors as good as possible.
Yes, look for a blue 1/2" PEX water pipe hanging down below the fresh water tank and open up the shutoff.