Regular towing near capacity with a 16+ year old tow vehicle may not be your best possible choice.
The 4L60E transmissions are not known to be the most durable.
The rest of the driveline and braking system in your 2004 has also aged.
Most likely, the wisest choice for heavy and regular towing, like you describe, would be to search for and purchase a tow vehicle better designed for the task.
If you insist upon only driving an Avalanche, a first generation 2500 series would make a much better choice for a tow vehicle than a 1500 series truck.
However, even though the 2500 Avalanche will be far more capable in terms of the weight being towed, the fuel mileage will be worse than what you are experiencing now.
Gas engines just aren't known for delivering the best fuel mileage when pressed.
Fuel mileage when towing is usually better with a diesel setup and the driveline hardware tends to also be more robust.
The right tool for the job is never a bad idea.