The grey tank is usually black plastic, so you can't see the liquid level through the sides. If you hook up your stinky slinky and pull the lever on the grey tank that would be the only inspection. If you can touch the tanks, rapping on the side at different levels can indicate level of liquid. Try it a few times under full and empty condition until you get an 'ear' for the different sounds. Knuckles or screwdriver handle is all that's needed, don't use an ice pick or nail hammer!
Tank level sensors are notoriously inaccurate, grey tanks can suffer from grease and soap scum build up on the tank sides. You already put soapy water in it, you could add a bunch of Dawn detergent to try to cut the scum and grease and let it soak, then drive a crooked road to slosh it around.