Pulling a Fifth Wheel problem, worried about my Dad...

Timbrens will had some stability when hooked up..........and are completely out of play (non issue) when not towing so overall ride is not affected.

BUT......

Really need to run it across the scales and see how much overweight combo is.
Over weight, rear suspension weakened, under inflated rear tires for load can cause that feeling of being pushed.

Properly matched trailer (weight) to truck (capability) a 5vr is very stable platform.

I think your 'check list' will get you an answer.