Actually it would be a low resistance in the motor (most likely caused by a shorted winding), but your problem will be in the motor. Only a motor or a short can cause too much current to be pulled. Since it runs for a while, it most likely isn't a short. If you run it on any speed other than high, you probably won't blow the fuse because on every speed other than high the current is limited by the blower motor resistor pack. That is what slows the speed. for now just lower the setting, that way you can probably limp through untill it warms up a bit. Make sure and listen that the motor is actually turning and not slowing down. If it stops, it can burn up the resistor pack.
It isn't all that hard to change. You do it from under the hood. IIRC there are three screws that hold it in. If you can get the other crap out of the way, it is fast.