The air flow direction on this vehicle is controlled by a mode door. Based on your selection, the mode door actuator changes the direction of air flow. What I would suggest is to remove the actuator and check it for operation. Chances are the actuator itself has failed.
Here are the directions for removal and replacement. The two attached pictures correlate with the directions.
1996 Chevy Truck K 2500 Truck 4WD V8-305 5.0L VIN M SFI
Actuator Removal and Installation
Vehicle Heating and Air Conditioning Air Door Actuator / Motor Service and Repair Procedures Actuator Removal and Installation
ACTUATOR REMOVAL AND INSTALLATION
Mode Actuator Location
Mode Actuator Replacement
Remove or Disconnect
1. Negative battery cable.
2. Electrical connector.
4. Adjuster links from control lever.
5. Mode actuator.
A. Electrical connector to the actuator.
B. Turn the ignition key to the run position and let the actuator position itself.
C. Turn off the ignition key.
D. Remove the actuator from the electrical connection.
Install or Connect
1. Mode actuator.
3. Adjuster links to control lever.
4. Electrical connector.
5. Negative battery cable.
Check circuit operation.
I don't know if you will find this helpful, but here is a link that shows how a blend door actuator is replaced. The process is very similar. The difference is the link discusses a different actuator in the HVAC system that is responsible for temperature and not air flow direction. However, they are appear and work very similarly.
Let me know if this helps or if you have other questions.
Images (Click to enlarge)
Thursday, March 26th, 2020 AT 7:30 PM