I have added a link to another site for Avalanches. It has all the info you could ever want to know about the towing capacities of the Avalanche.
WHAT IS TOW HAUL?
Avalanche comes with a four-speed automatic transmission with overdrive and Tow/Haul mode. When activated, Tow/Haul mode reprograms the automatic transmission to reduce the frequency of gear shifts when towing. SOURCE: The General.
WHERE IS MY WIRNING HARNESS FOR MY BRAKE CONTROLLER?
Chevrolet no longer provides you with a wiring harness for you brake controller. They do not provide the 40 amp fuse or the 7 to 4 wire adapter for the wiring connector. 2003 model.
SO HOW MUCH CAN I TOW?
Trailer towing 1500 2WD 1500 4x4 2500 4x4
3.73 axle ratio (lbs.) 7200 7000 9,900
4.10 axle ratio (lbs.) 8200 7900 11,900
Maximum trailer weight ratings are calculated assuming a base vehicle, except for any option(s) necessary to achieve the rating, plus driver. The weight of other optional equipment, passengers and cargo will reduce the maximum trailer weight your vehicle can tow.
SOURCE: Then General.
Here are some general rules of thumb, these have been based on the Insurance industry reviews after accidents, as with all rules of thumb your situation may not apply but they are a good reference.
Tongue Weight Rule of Thumb
Tongue weight = 10% to 15% of actual loaded trailer weight.
Trailer Weight Rule of Thumb
Maximum trailer weight being towed...should only be 75% of the weight rating of your tow vehicle. (CGWR being taken into consideration). If factory rating is 5,000 lbs. trailer,... a 3750 lb. trailer weight is recommended, etc. If 7,000 lb. trailer is stated,...a 5250 lbs. trailer weight is recommended, etc.
Weight Distribution Hitch Rule of Thumb.
When tongue weight approaches 350 lbs.,...a weight distribution hitch system is recommended.
Wheel Base Rule of Thumb
A tow vehicle should have a minimum wheel base of 100 inches. A wheel base of 110 inches is recommended for a 20 foot trailer. For every additional foot of trailer length (above 20') add 4 inches to the wheel base. i.e., a 22 foot trailer=118 inches wheel base, a 24 foot trailer=126 inches wheel base. Etc. However, when you get into the larger, full size tow vehicles, such as Av's, Surburban's, F-150's, Ram Trucks, etc. this rule relaxes, as these types of vehicles (if properly set up) are capable of towing much larger trailers than this rule would suggest.