OK, and I know I will take grief for this, but, using your manual shift control is necessary for some conditions.
I have the exact set up (2010 Ram 2500 with the diesel and automatic transmission). I tow a CC Silverback with a dry weight of 11K + lbs.
Where I live, in northern Arizona, there are numerous hills to climb. I should mention that I have both turbo and transmission sensors and closely monitor the temperature of both.
When going up these hills (3 to 6 % grades), I will generally shift down to 5th gear. This, I have found, drops both the turbo and transmission temperatures. I hear what others are saying ...let the automatic function do the work, however, in my experience, it want always drop into a lower gear, which results in the turbo getting above 1,000 degrees, which I try to avoid.
Bottom line, it want stress the transmission either way. I just like being in control of things I can.