I have a 2012 Chev 1500 5.3 3.42 6 spd/ Trailer is 6600 loaded. Tongue is heavy 1200+. From Ont Canada we have made round trips to the maritime provinces, Florida, Los Angeles, San Diego and we're currently in Quartzite AZ. The truck has handled the trailer with no problems. I really like the Equal-i-Zer for stability.
Yes the engine does rev on the long steep slopes but it does not hurt the engine. It keeps speed all the way.
You will find the newer truck a big change from your 2000.
From my experience you will be fine with your plan.