Unless it has been retrofit by a previous owner, you only have one tank. Grey (wash water) tanks were not "standard" until 1973-4.
That being said, here's what we do when camping with hook-ups:
Our drain has a "Y" which allows the grey water to bypass the tank. So we leave the sewer hose attached for the grey water to flow all the time. We do leave the black tank valve closed though to keep the liquids in with the solids. When the valve is left open, the liquids drain leaving the solids to dry out and possibly become "stuck" either in the tank or in the hose...either is yucky! When the tank is full or has some volume, then fill it with water and open the valve to clean it out. Then follow with a fresh tank of water. We leave about a gallon of water in the tank to "keep things floating" so you don't get a "cone" developing. We usually do this every day or every other day. We do add chemicals in between dumps...but a lot of people don't like to. If your valve will not hold a seal thus all your liquids draining out, even when cleaned & closed, it may be time for a valve re-build.
Before leaving or storing, we usually close the valve, refill with water & dump again 1 or 2 more times. So it's good & clean.
If you have some residue build-up
it's a good idea to add a bag of ice to the empty tank right as you're getting ready to travel, the rough ice and the movement of the trailer scrapes the inside of the tank, then you can empty it until arrival at your next stop.
Each trailer model/year plumbing is a little different so someone with the same model/year may be able to give you better specifics on your valves & bypass configuration.
There are lots of threads in the plumbing section that can give you other people's routines or info about chemicals & cleaning/rebuilding the valve in the meantime.
<edit> looks like they beat me to the punch...basically same info confirmed.