I've been towing our 5000lbs loaded HTT for over 10 years now.
The Avy is a Suburban with a bed. So it has lower payloads than a Silverado.
That 8100lbs amount is for an Avy with 4.10 rear end, which wasn't common.
My 2007 has the factory tow package, towing max of 7200lbs and a payload capacity of 1357lbs.
You need to post your Avalanche's specs and payload capacity.
Where did you get the trailer'so 6000lbs weight?
Also get a WDH with sway control to tow with, NOT just a sway bar?
And the dealer did you a multitude of wrongs. Letting you drive off without a brake controller and a WDH was a biggie.
And never believe anything a dealer tells you without checking it first.