Trailer hitch setup

Look at the tires, are they under inflated?? Inflate they to the tire max inflation. Are they 4 ply instead of 6 or 10 ply??

FYI, here's a link to another thread where someone was having a similar issue as you. Maybe something said on this thread will help point you in the right direction. This was fairly recent. Click here.

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 ?

Don't know if this will help, you've probably already seen it. Hope you followed all the guidelines: Towing guide for 2016 Expedition

Wonder if the 2016 model is simply more light weight than your 2005 model?

Tires could be under inflated.

You might have to shift some weight in the camper further forward (tail heavy)?

Did your new Expedition come equipped with the factory installed tow package or did you simply add a hitch? If it was not factory, it's possible the suspension may not be adequate for the weight of the trailer you're towing. (I don't know how Ford does this).

I'm just grasping as straws here, but from your description it just sounds like the tow vehicle suspension is too light.