You don't mention your engine size or transmission gear ratio. But.. We had an F150 with a 5.4L Triton V8 and 373 gears. Our TT then had a 7500# GVW. We pulled for 3 years throughout the Rocky Mountains. You will be shifting a lot both uphill and down, but that's normal. Your 2500 should be equipped with "tow/haul" mode. Use it. Plan on seeing 3500 to 4k rpms keeping your truck in it's best power band. All should be good in my opinion. Key to maintaining control downhill is to start slowing early. Down shift early and use the marked speed limits for big trucks as your guide.
Feel tree to PM me if you'd like any specific info on the parks or camping in our "Pretty Great State"
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