Glad you got it fixed! I had the same truck with the 8.1 (but same trans, 4L80E) and it would run at 175* but never got hotter than 200* no matter the hill or outside temp. Our TT was close to 9000 lbs loaded.
The transmission actually has 2 coolers, the cooling line goes from the trans into the right side of the radiator, then out of the radiator and into an external cooler mounted in front of the a/c condenser, then back to the trans. The cooling was very effective and never had the need for an aftermarket cooler.
Not sure about the 6.0L, but the other side of the rad on the 8.1 had an engine oil cooler. That was a fun job replacing that, dealing with oil, trans fluid and coolant in the same bucket :)
Anyway, now that you have replaced the rad, I would drop the pan and change the filter in the trans at the next fluid change. These came from the factory with Dex III fluid, the new GM fluid is Dex VI, which is full synthetic and backwards compatible with Dex III. Of course there are different opinions about using Dex VI in an older vehicle, but I found the full synthetic ran cooler than the Dex III.
