I think you are looking at the connector wrong. The ground terminal is on one side, not the center. Looking at the socket, with the two parallel tabs on each side and the other tab on top, the ground should be on the right. (This isn't my photo, just one I found)
Since the low beam terminal will be grounded through the other headlight when the high beams are on, I think what you found is perfectly normal.
As mentioned, I think the problem is a high resistance connection. Try sliding the connector onto the bulb partway, and then checking for voltage directly on the bulb pins. I'll bet you find that the voltage "goes away" when the bulb is connected, indicating a bad connection somewhere up the line. And since it is affecting both beams, it's almost certainly the ground connection. You can double-check that, by measuring between the bulb pin and a good ground (like the battery negative post). As TOC suggests, running a new ground wire back to a solid ground point is an excellent idea.