If you have sewer hook-up where you are going, you can treat your situation like a trailer which has no gray tank, like our '65 Caravel
. Hook up to the sewer, your gray water will always drain. When your black tank gets nearly full, dump it. Some may back into your gray tank, maybe not, depending on how things are set up. In any event, after dumping your black, close the black valve, and run several gallons of water into your gray tank to wash things out.
It would still be best to change the valve on your gray tank.