Usually, there's a resistor block that fails. What they do is:
On "Hi", full voltage (say, 12v although it's actually 13.8 or so while car is running) goes to the blower.
on the other settings, the switch routes the voltage through a resistor. So the motor runs slower. It takes usually, about the same amperage, regardless of setting.
The resisitors will break/ go open causing what you describe. Usually, the resistor is located near the output of the blower motor, in a duct, that is accessable under the passenter side. Sometimes, you need to remove the glove box. A shop book should show you where. Replacements run $10-30, depending on vehicle. Try that next.
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