From personal experience with my grey water getting sluggish this summer.
You've been advised to NOT leave the black tank dump open all the time. That leads to the solids mounding up and not being carried away by the liquids. Black tanks must be dumped in a whoosh so the liquids carry away the solids.
Grey water, by comparison, is very "clean." The solids are small. But nothing says the grey tank...constantly dry...can't build up a layer of solids that can slow or block the drain.
You say your grey tank is full but won't drain. I'll start with a premise and expand from there. Unlike your black tank which may have a black tank flush, and, if nothing else, has a toilet through which you can feed a hose and inject lots of water at city water pressure, your grey tank has no such flushing mechanism.
A suggestion that may get a bit messy assuming your grey tank is "full."
Get a dump adaper that has a water injector and gate valve so you can push water up into the tank.
Have a "dirty" working garden hose...not your drinking water hose.
Leave the black tank gate valve CLOSED.
Open the gate valve on the grey tank. Grey water should fill the clear adapter.
Keep the gate valve on the adapter closed.
Use the hose adapter to inject a SMALL AMOUNT (perhaps 1 or 2 gallons) of water up into the grey tank.
Have a helper inside the rig monitoring your kitchen and bathroom sinks to see if they begin filling with grey water.
Shut off the garden hose water injector.
Open the gate valve on the adapter and see if grey water dumps out of your grey tank.
Some may dump and then dried clumps of garbage (food particles and coffee grounds and such) may once again slow the draining. So repeat the process a few times until the grey tank dumps with a whoosh.
Next, once you are sure your grey tank is draining, bring your garden hose inside.
Be sure it has a shutoff adapter on the end. Point the garden hose straight down the kitchen sink, bathroom sink, and shower/tub drains and run lots of water down the drains into the grey tank to help flush out remaining solids on the tank floor.
Next, the black tank.
Turn off the city water to the rig. With the black tank gate valve closed, use your dirty garden hose straight down the toilet until the black tank is about half full.
Add a dose of tank treatment. Let it sit and do its job for 24 to 48 hours BEFORE YOU DUMP. If you only fill the black tank half way, you'll have plenty of room to use the toilet for a couple of days.
Next, empty the black tank. While dumping, monitor the effluent through the clear adapter. If hard chunks seem to be blocking the dump, inject a bit of water through the adapter to push the chunks back into the black tank and repeat the "softening" process for a couple more days, then try again.
The condition of your black tank may be somewhat critical. You MUST keep the gate valve on the black tank closed until the tank is nearly full, and then dump. This carries away the solids rather than letting them build up and dry on the tank floor...A.K.A. "mounding." Then always "chase" the black tank dump with the dirty garden hose down the toilet. But the same thing, to a much lesser degree, can happen in your grey tank. Leave the grey water in the tank to keep solids from drying on the tank floor. Water in the tank will whoosh away the solids, and once in a while you can flush out your drain pipes and the grey tank with the dirty garden hose down the sink/tub drains.
RV plumbing is NOTHING like home plumbing. When you flush at home, about 1.5 gallons of water chase the waste down the drain. And, with your kitchen/bath/shower drains, the volume of water you use at home is typically much higher. And in both cases, there isn't a 30 to 80 gallon holding tank in the lines to give a place for solids to build up and dry.
That adapter I recommended allows you to inject water "backwards" into the dump plumbing and tanks, allowing you to push solids away from the dump opening and get things moving.
Sorry for the short book, but NEVER leave the gate valves open, or solids build up in both tanks. Plus grey water is the ideal thing to use to chase black water down your stinky slinky (or other). Tissue and other solids in black water want to cling to the stinky slinky, but the grey water flush out leaves nothing but a few coffee grounds.