Well, long story short… I ran a new control wire from the one valve box I could find back to the controller and it is working now. I feel like I got a little lucky here in that the rest of the stations don’t have some kind of issue with this, but I guess now that there is a path back for the current to flow, all of them can make it.
It may just be my lack of paying attention in high school physics, but I still don’t understand how the voltage potentials could be 28VAC everywhere, but no current was flowing.
Thanks @Thomas_Lerman and @DLane for your support!