Normally adding weight to the rear should make the front less responsive to steering, but as you say a full gas tank improves handling, it might be that just adding weight is helping your situation.
Try filling the water tank and see if that helps. I know the conventional wisdom is to travel light with empty tanks, but with your rig, you should have a lot of safety margin in your GVWR, and it might help balance the load.
Since your water tank is just about as large as your fuel tank, it might help if you fill it... depending of course where it is located on your chassis.
I know what that feels like when cornering. I used to tow a boat that weighed more than the pickup. When I went around corners I had to really slow down as the boat would want to push the rear of the truck forward.
It's just a matter of vehicle weight vs. trailer weight.
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