Some other thoughts:
I assume the weights are from going to a scale?
The Jayco specs say the CCC is 1040 lbs. That's pretty low for a 31' TT. UVW is listed as 5960 lbs and GVWR is 7K lbs. The weight of all cargo for camping, factory options and mods/upgrades is typically (anecdotally) around 1,000 lbs but can be more and sometimes less. Is it possible you're over the GVWR?
Tongue weight is normally in the 12-13% range making your actual TW likely to be around 900 lbs. Can be up to 15% in some cases, like on our 29' TT. Are the trunnion bars rated high enough? We have a Reese DC WDH and Reese defines their rating as cargo wt. + tongue wt. which is different than all the other manufacturers.
What is the payload capacity of the truck? If you've gone to a scale, just take the wt. of the truck and subtract it from the GVWR on the door pillar sticker. Actual payload can vary quite a bit in some cases. What is the wt. of all payload in the truck when fully loaded for camping (passengers, pets, groceries, camping stuff, full tank of fuel)? How does the actual payload cap. of the truck compare to the payload + TW?
What tires are on the truck? If P rated, replace them with LT type.
Tire pressure on the TT tires should be at max sidewall psi. PSI on the truck tires should be inflated to or near max. I run our LRE LTs at 75 front and 80 rear.
Are you towing with one or more full holding tanks? Not only could that put you over the GVWR, it could affect the TW a lot depending on location of tank(s).
You can also adjust a WDH at a scale instead of using the fender height method and is more accurate.
Is it possible the front end needs a rebuild? I had to do a complete rebuild on ours at only 3 years old and about 100K miles. It was a lease return and had been used in an area full of potholes and rough roads.
Heavy duty shocks on the TV can help a lot. I installed Bilsteins and they really made a difference. Shocks on a TT can also help a lot but are not a simple bolt-on project. We added them to our TT and they made a HUGE improvement in sway and handling.
Is there any play between the WDH and receiver? Reese has shims available to remove excess play. I installed them on our WDH.
Sometimes it can be multiple things that need to be addressed to get it to where it should be. Posting a pic or two of your TV & TT hooked up may or may not help. Once you do get it figured out, towing should be a breeze at any speed (up to rating of the tires of course).
