It sounds like a light tongue weight issue . However, if all is the same as before and all you have done is swapped TV's , then maybe the suspension has a change to it from 2005 to 2016.
Just to go over the basics:
Check the air pressure in the TT tires as well as the truck. max it out as you test things.
Without hitching up , measure the trucks rear height as well as the front height. ( I use my wheel wells)
Now hitch up the trailer to the truck w/o using the WDH and measure the rear/front heights using the same reference point to measure. See how much the tongue weight has made the trucks rear drop while lifting the front.
Now adjust the WDH to match those original numbers making it level. The truck should now equal the original measurements, making it look level as well as the TT. IF, the TT does appear to be level now, then you need to readjust the hitch height to match this new truck.
At this point, if you do not notice any better handling of your rig, then something in the new truck suspension is changing the handling OR something has changed in the TT, either by transferring weight to the rear of the trailer or by diminishing the TT tongue weight .
EX: did you remove two batteries and install just one or maybe add a spare tire to the rear bumper ? Are you carrying something on the rear that is now adding weight ?
Are you using the same size and rated tires on the new 2016 truck ? Could the sidewall on the new tires be taller, causing motion (flex) that you didn't have before ?