[joe231]

Yes both wires will read hot to a seperate ground at that plug as long as the circuit is open, the hot will feed through the bulb. It is all about potential difference. When you turn the switch and it puts the white wire to groung, it actually completes the circuit (or loop, as you put it).
If the wire coming out of your headlight is a good ground, then your problem is certainly the white wire between the headlight switch and the plug by your seat.

Be sure to check the contacts in the plug. You can also use your meter set on ohms. Put one lead on the white wire at the headlight switch plug, and the other on the white wire at the plug by your seat (with the plug unplugged). You should have extremely low to no resistance.

You can do a voltage drop test on it also, I can explain that if needed.
I don't know your electrical experience, or tools you have.