You might examine the water supply line behind the toilet (water supply for flushing the toilet) to see if that's leaking / the source of water into the shower pan and thence filling the gray tank. Or perhaps one of the hot / cold supply lines under the sink in the bath is leaking.
IIRC you are at an extended-stay situation and connected to city water, do you have a pressure regulator installed between the park water connection and your trailer city water port? If not, that might be a good idea - ask if other folks in the park use a regulator on their water connection, and if yes they can show you what and how to install that.
I can't imagine anything relating to the rooftop AC that would bear on this.