Gotta ask, just in case, did you check BOTH fuses? One in the under hood power distribution center and one in the mini box. Takes both for the blower controls.
Anyhow, looking at the wiring diagram, your relay on a 99 is a 4 terminal.
Power from both fuses to two of the terminals, a ground to another and the power feed out to the blower motor from the forth.
Similar to a standard 4 terminal Bosch relay. Power in to run the blower, power for the trigger, a ground, and power out to the blower.
From the looks of it the relay is hot all the time with the ignition on. Again to the motor, not the switch.
The resistor (springs) control the ground to the motor via the switch to control speeds. The switch just makes the connection to the various resistance springs to a direct ground.
The switch gets it's ground connection from the mode switch.
Only place I spotted a power lead going to the control head was for illumination and to work the blend door.
Hopefully this link works, you'll want page 8w-42-3 for the blower motor/controls schematics. Give it a bit, it takes a little time as it's loading all the wiring diagrams.
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