This must have been a fairly common issue. A quick Google found several referencing the same issue and here is a copy of a GM service bulletin. I found this in a Duramx forum where a guy was describing your same symptoms.
SPECIAL COVERAGE
Bulletin No.: 11046
Date: April 07, 2011
Subject: 11046 - Special Coverage Adjustment - Heating, Ventilation, and Air Conditioning Blower Not Fully Functional on All Blower Speeds, Inoperative, or Runs Continuously with the Ignition Off
Models:
2003-2006 Chevrolet Avalanche, Suburban, Tahoe
2003-2007 Chevrolet Silverado
2003-2006 GMC Yukon, Yukon XL
2003-2007 GMC Sierra
Equipped with Manual HVAC (CJ3) or Heavy Duty Heater (C42)
Due to part availability, owner letters will be released in phases starting with older model year first. Owners for model year vehicles not included in first phase owner mailing will receive an owner advisory letter.
The letter will inform the customer that parts are not yet available and that another letter will be sent to them when parts become available. A copy of this letter will be attached to the dealer message announcing this bulletin.
In the meantime, if a customer brings their vehicle to the dealership with the condition described in this bulletin, repair the vehicle using the parts called out in this bulletin.
Condition
On some 2003-2006 model year Chevrolet Avalanche, Suburban, Tahoe; GMC Yukon, Yukon XL; and 2003-2007 model year Chevrolet Silverado and GMC Sierra vehicles equipped with a manual heating, ventilation, and air conditioning (HVAC) system (CJ3) or heavy duty heater (C42), the interface between the electrical terminals of the relay resistor module and the wiring connector that powers the module may be incapable of conducting higher current levels for sustained periods of blower motor operation. In addition, moisture and other contaminants may enter the fresh air intake plenum and contact the internal circuit of the module or corrode the terminals. Either of the above may cause the relay resistor module or wiring connector to overheat, resulting in one or more of the following symptoms:
- The HVAC blower may not function on certain or all blower speed settings
- A burning plastic smell or smoke may be present in the vehicle
- The HVAC blower may run continually with the ignition in the OFF position
In very rare events, if the above symptoms are ignored, a fire could occur.
Special Coverage Adjustment
This special coverage covers the condition described above for a period of 10 years or 240,000 km, whichever occurs first, from the date the vehicle was originally placed in service, regardless of ownership.
Dealers are to install a new blower motor resistor and resistor module connector. The repairs will be made at no charge to the customer.
For vehicles covered by Vehicle Service Contracts, all eligible claims with repair orders on or after April 8, 2011 are covered by this special coverage and must be submitted using the labor operation codes provided with this bulletin. Claims with repair orders prior to April 8, 2011 must be submitted to the Service Contract provider.
Vehicles Involved
Involved are certain 2003-2006 model year Chevrolet Avalanche, Suburban, Tahoe; GMC Yukon, Yukon XL; and 2003-2007 model year Chevrolet Silverado and GMC Sierra vehicles equipped with a manual HVAC (CJ3) or heavy duty heater (C42).
