Jane, I reread this whole thread. I can't imagine how frightening your experience must have been, and unless I could find and fix the cause, I would not tow with this combination again, either. You have a substantial amount of money invested in the trailer and vehicle (sweet!!!), so it may be time to involve a professional hitch specialist to help you figure out what went wrong and get it set up right. I have to believe it can be fixed, and I am thankful nothing worse happened so that you have another chance!
It seems to me there are two possibilities: either trailer sway is causing the ESC to activate (most likely), or the ESC is activating for some other reason and causing the sway.
Taking the second possibility first, if the ESC is initiating the sway condition, I can think of several possible reasons. (1) The WDH was improperly adjusted and allowing some wheel slippage on an inside wheel during cornering (at the front if under-tensioned or at the back if over-tensioned). (2) The trailer brakes and/or controller are not adjusted correctly, causing an unexpected tug or jerk on the vehicle and activating the ESC (but you said you didn't touch the brakes, before or during? the incident… hmmm…). (3) The ESC is malfunctioning or improperly set for towing, as Vic suggests. If you haven't already, a careful read-through of everything in your owner's manual about the ESC system and towing would be a good start. Your Durango R/T was designed as a high-performance vehicle, so the ESC may be programmed differently from the run-of-the-mill systems in other vehicles. It wouldn't hurt to have a conversation with a Dodge technical specialist. If they don't return calls, a letter might be in order.
On the other hand, if sway is the initiating factor, it has to have a cause. A number of them have been mentioned: low tongue
weight, nose-high attitude, excessive speed, wrong tire type or low pressure, shifting
weight (water sloshing in a half-filled tank, for example), improper cargo loading, faulty trailer design, to name a few.
If it were me, I would start with some baseline data regarding the tongue and
axle weights of this trailer, preferably loaded for travel. I would definitely want to understand more about the ESC and towing systems on this specialty vehicle. If the brake controller is built-in, I'd want to understand how it works in tandem with the ESC system. In a sway situation, you really don't want vehicle braking without trailer braking. And I would seriously consider working with a hitch specialist in your area to choose and set up the WDH.
Best wishes, and don't quit just yet!