Curiously, your tanks may be easier to clean effectively if you wait until they’re fairly full of liquids. Gravity can be your friend when flushing out the black tank, and any floating organic bits in your grey tank have a better chance of clearing if the water volume is high.
No matter what you do, don’t let your black tank get dry. This is when things get, um, sticky. Dried solid waste won’t go anywhere, and constipation isn’t good for you or your RV. Once you dump your black tank, add water back immediately, drive for a time to agitate, and dump again to make sure there’s nothing building up or sticking to the walls of the tank.