Gray tank plugged, limited options

Running water backwards into the tank might loosen the clog, but then it will start pushing water up your drains. Could make a huge mess if you aren't careful. Doesn't sound right that running water in the sink ends up on the ground. I would start with the camera and see what you can by running it up the drain connection. Then maybe make a small hole for the camera near where the water is coming out and try to see why that is happening. My grey tank has 3 connections. Waste in, waste out, and vent which goes through the roof. There is no place where it could overflow onto the ground unless I somehow pushed water up the vent and it ran out onto the roof.