Buy one of these, Valterra T58 Twist-On Waste Valve,
Twist-on Valve - Valterra T58 - Sewer Fittings & Adapters - Camping World . Install it on your sewer dump fitting and
close it. When the grey tanks fills, go outside and first open the gray dump valve, then open the black dump valve. The grey will flow into the black tank and I doubt the grey tank becomes very contaminated. After the transfer be sure to close both the grey and black valves. When I do this the water level is then equal in both tanks. The second time you transfer grey to black the efficiency is not as good as the first. After 3 times you better go to the dump.