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I have a 91 F150 and when I turn the blower on, I see the volt meter drop alot. I put a volt meter on the battery just to confirm and I can see the voltage drop and continue until the battery almost is dead. I had the alternator checked as well as the battery and they are fine. Took the blower and attached it directly to the battery and see the same condition. The blower works fine and the fuse for it has never blown. Does it sound like a blower needs replaced?
If the motor is pulling so little current that it DOESN'T blow its fuse, it should take about 5 hours with the engine off to kill the battery. If it dies just in the time you're sitting there, then the battery IS bad. The motor is bad, too, if it's not spinning, but it would catch fire if you used a stopped motor to kill a good battery.
Maybe your alternator is not putting out enough current. Try running at a fast idle and check the battery voltage. Then turn the lights on and see if the voltage stays around 14v. Then turn on the blower motor. They draw a fair amount of current, 10 to 20 amps.
With the headlights on and the engine running, a voltmeter on the battery shows a little over 14 volts. When I turn the blower on high, it first drops to 12.5 and then starts dropping a tenth of a volt untill I turn the blower back off. The voltage then climbs back up to a little over 14 volts.
Get a Haynes manual and look in the wiring diagrams in the back to find the dash ground. The ALT light is part of the control for the alt's internal voltage regulator, so if the ground is loose, it might behave that way when the blower fires up.