I'd imagine towing a trailer with any sort of significant tongue weight would have a huge effect. Yours is only a few feet shorter than mine....and mine has a long overhang for sure!
The rear axle on these things are heavy.... mine is at the limit while the steer axle is under. Put on a few hundred pounds of tongue weight and it'll lighten the steer axle even more while loading the drive axle more too.
I've not towed with mine yet either, but I'll say that after I had the alignment done and after I set the tire pressures based on load I saw a noticeable improvement..so I never went forward with any of the common mods or after market things. I think getting the tire pressure's right was key.
For my rig and what I do experience, I think the solution that most directly points to the issue is a panhard rod. I'd guess that something being towed....the tail wagging the dog.... would just add to that issue...and that is a direct result of the leaf spring suspension.
here's a decent little write up on the various solutions...and this is describes well why I don't think the steering devices, such as the safe-t-steer are the most direct fix (for mine anyway). No doubt they help, and I'd like to have one, but their primary intent is for something else.
I think another big consideration regarding handling, and I'm not thinking this is an issue here really for the OP, but for others....and that is driver expectation and attitude. My wife doesn't want to drive the thing...she's intimidated by the size of it...and that sets the stage for bucking her expectation that it's supposed to drive like her Buick. I just think it helps to remember that you are driving what amounts to be an overloaded medium duty truck.