I have been chatting with a few other experts on this one and I got to say that we are somewhat confused by all of this. The 10 volts you are getting at the blower plug should be enough to operate the motor. As fas as the resistor goes, its out under the hood on the heater case. Here is what must be done to replace this unit.
Cut all four sides of the cover in order to remove the three attaching screws. Use a sharp utility knife in order to cut the case. Make the cuts straight and clean because you must reinstall the access cover and the access cover must be visible to the customer.
blower on high 0 ohms
blower on low no connection
No fuses under hood, instrument panel fuse is good.
You won't show the short on low as the resistor is not plugged in. What really is making me wonder is that you verified 10 volts then 12 volts at the blower motor connector. If this is the case and this voltage is being shorted, It should be blowing the fusible link (I did mispeak, this does have a fusible link rather than a fuse for the high speed). It may be worth a shot to verify if you do in fact have voltage at the red wire of the high speed blower relay. Here is a better illustration of this.
Here is where the relay is located.
yes I have power at the red wire of the relay. nothing else has power.
At the relay the purple wire that goes to the blower has no connection if unplugged from motor and 0 ohms if motor is plugged in.
I hate electricity!!!
OK wrapping up for the night.
Put everything back together, plugged everything back in and guess what worked. Yup, blower worked. I have no idea why. I am afraid by jiggling some wires things are working (did I mention I hate electricity). I think maybe by messing with the relay. I am planning on replacing the rely for the sake of doing it. What do you think?