Sway bar is broken, or the rubber bushings are worn out, the mounts for the bushings has come loose, or the end links are broken/disconnected??
Photos would help if posted up.
The wetness appears to be from the rear pinion seal. Check the breather on the rear axle to see if it's functioning. Push/pull on the driveshaft (side to side) near the rear diff to check for axial play (should be none). IME, you may be able to just replace the rear pinion seal if nothing else...