The only way the fan control module will operate the fan is when the module understands the commands from the PCM for the desired fan speed. Simply grounding the white wire tells the module to set fan speed at 94%. The module then controls the fan motor speed. In your case any of the modules you have are not translating the correct fan speed and turning on the fan (and or setting codes). Both P0480 and P0691 are cooling fan speed output circuit problems.
Cooling Fan Speed Output Circuit
The ECM detects 0 voltage on the output during commanded transitions.
Cooling Fan Speed Output Circuit Low Voltage
The ECM detects low voltage on the output while commanded OFF.
I moved some components around in the diagram I sent last night and is in the link below. Basically I moved the fan motor to show it separate from the module. The round 3 wire connector is to the module and the 2 wire terminal is obviously to the fan.
Cooling Fan Assembly Circuit
In the first diagram I sent GM shows the Engine Cooling Fan as a unit, so much so that the only way to get a fan motor, module or even a fan blade from GM comes as a complete assembly. The aftermarket world, forums, and all the other riff raff are basically trying to shortcut and copy what GM has specifically designed to operate properly for this vehicle. I used to read through forums years ago, but have found the information I pay for each month is much more reliable and provides me with correct diagnostic procedures.
I'll be hear most all of today, let me know if you have any questions.