Does your new Avalanche have the:
5.3 L Vortec 5300 V8 with Active Fuel Management, 320 hp (239 kW)
6.0 L VortecMAX V8 with VVT and Active Fuel Management, 366 hp (265 kW) (late 2006)
I had a 2000 Chev s-10 Xtreme with the 4.3L engine and pulled a 3000 lb Boat & trailer and it pulled O.K. - Then I got a 2004 I6 trailblazer (275 H.P) and it pulled the same boat much better, a bit more torque then the 4.3L V6 (180 H.P) . The trailblazer weighs in at 5,750 lbs. The S10 xtreme was around 4000 lbs..
So I would think that going from your older blazer at around 4000 lbs and 180 H.P & 245 ft/lbs of torque to around double the H.P (320 - 366 H.P) and 5700 lbs of curb weight it should pull great