Filling the water tank, at 8.4 pounds/gallon, could be a very unsafe idea. What if the twitchiness is caused by already being near weight capacity or a weight distribution issue? Where is the water tank? If it's under the bed, in the rear of the RV, that's the last place you want excess weight since it's already stated the RV is stable without the toad.
This is a Ford F-450 chassis, making me think it's a Class C. The front wheels are in front of the driver, the rear axle at the chassis designed spot, unless the frame was extended. Probably a long distance with frame extensions from the rear axle to the receiver. A bit of slop in the receiver would make annoying clunks, but not cause the large changes in handling the OP reports. The amount of motion needed to create these changes would be far more than I've ever seen in receiver/draw bar set up. I doubt if tightening the slop in the connection will do more than quiet the noise down and empty the wallet.
Get the RV weighed, making sure the rear axle capacity isn't exceeded, set tire pressures per tire maker recommendations. Watch tires for improper wear patterns indicating alignment issues, or get it aligned, but I'd assume it was done when the stabilizer and sway bars were installed.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI