98 chev K1500, 350, auto. Usually towing a lighter trailer, 1500+ lbs.
My manual says to tow in OD unless the tranny is "hunting".
I stay in overdrive until the tranny makes its first downshift on a hill. It may just be unlocking the torque converter, I can't tell. As soon as it downshifts and the hill is reasonably long I bump it down into 3rd and cruise until the terrain flattens out and then I bump it back into OD. I don't tow above say 65 so 3rd isn't screaming, maybe 2500 rpm. I just don't like the revs high unless they are needed and I don't like to speed up and slow down just to prevent downshifts. My method keeps the truck speed constant and allows the tranny time to cool between shifts. If someone thinks shifting on the fly is a problem, please speak up.
As long as there are several minutes between shifts, your tranny should be fine. Hunting causes the excessive heat and hot spots that ruin the tranny. All tow vehicles with an AT should have a tranny temp gauge.
If this common debate leaves you with no conclusion, just do what the manual says and slow down until the revs are acceptable. Just think, if you buy bigger tires the revs will be lower in 3rd while towing, is that a good excuse or what?