Dumping gray water on the ground is fine in some locations, but not at a campground. For example, it is allowed for dispersed camping (boondocking) in many National Forests so long as you aren't doing it within a certain distance of a body of water, campground, etc. There are specific rules that vary by location and managing agency.
How large is your black tank? Maybe you could install plumbing to run the gray water there. That's the way my TC is. I'd like to have a valve to divert gray water to the ground when possible, so may install one of those at some point.