Grey waste tanks have vent lines.....they are what sticks thru roof.
Air admittance valves are installed after sink 'P' traps so that when draining grey tank the 'P' traps don't get sucked dry. The AAVs open when draining allowing air into sink drain lines after the 'P' traps.
If the stick closed (very unusual----usually stick open) water would gulp-gulp-glup at sink
Water backing up into shower when draining sink.........then as suggested...the drain line AFTER shower is clogged/obstructed and not allowing full flow into grey tank.
Need to get access to shower 'P' trap..remove it and then snake drain line from there down into grey tank.