1) Why are there two control boards? One board powers a set of two crossbucks.
2) Ditch the extra, external 1k resistor. Someone may have been using that board for another use.
3) "The signals work just great if I apply between 5V - 8VDC." I am assuming this is directly to the signals themselves. There is a 15VDC regulator after a half-wave rectified supply, on the control board. That's why the instructions show a minimum of 14VAC input to the board. (The rectifier supplies a higher voltage to the regulator)
4) "When I tested the output end of the resistor I got 0 volts." Tested to what? The WHT wire is the output from the 15V regulator. The GRN wires sink the LED current through NPN and PNP transistors (when active), alternately driven by a 555 timer, as the crossbuck flashes.
As Stan said, just hook it up to a 14VAC accessory post, as shown.
Dave