You don't say how big your chicken run is, but chickens will kill all green stuff in a run and dig holes for dust baths. I doubt it is just your geese causing that problem.
The only way to fix a problem like this is to give them a tremendous amount of space, and, as Olive Hill said, they will still destroy the green stuff and dig holes in high traffic areas, whether geese or chickens.
Keeping ducks or geese with chickens is a little challenging because of the water issue. It's just the way geese and ducks drink and handle the water. They will make a mess. Keep your water out of the coop and arrange for good drainage around the waterer in the run. Building up pea gravel or sand in that area will help, but unless you are built on sand, the water needs some place to drain to. I can't tell you how to manage that.
Good luck!