On a LWB HD truck there is nothing wrong with not returning all the weight to the front axle, in fact everything I have seen shows the OP as correct with about 50% restoration. Good luck trying to get 100% back, you will break things on a heavy TT.
It sounds like a road issue, but your shocks are probably worn out.
I would also hit the scales and check your axle weights and set tire pressure based on actual weight and not door placard. The door placard is based on FAWR/RAWR weights, not actual loading.
10% seems very low for a TH, even 13% seems low for a TH unless the garage is FULL. Do you weight the TW only each trip? Or the TAW too?
Have you tried bumping it to 15-16%. More TW will put more weight on the TV and might stabilize things out.
Also, does Hensly make an extended shank? It may be the position of the TV axles to TH axles and increasing it by 6" might do wonders. I know my TT tows much better on rough roads with the 6" longer hitch.
Lastly, what condition are the trailer tires in? Proper pressure?
