Time to up-truck. You have a HD truck.. just not enough HD truck.
Band-aids will never ever ever ever change the fact that you are likely very overloaded.
If you are going to play with a heavy fiver, fact of the matter is that you need a truck to carry the load, not just pull the load.
So, I looked up your trailer the 2010 Jayco Eagle 351rlsa, that trailer is 11050 DRY and has a GVWR of 14950 lbs... so your pin weight is between 2762.50 lbs to 3737.50 lbs and just taking the medium it would be about 3250 lbs ish. Plus 250 for hitch, plus 300 lbs for passengers, and your are seriously looking at close to 4000 lbs you are putting on your truck. That is one ton dually range.
Ummm. thats a LOT of weight for a lot of HD trucks and More than most but the most basic 3/4 tons are rated for.
So, my recommendation... up-truck...