Measure the D-cell voltage between the ends of the D cell, not from one end to ground - they are not grounded at either end. New cells will run around 1.56 volts. When they are down to 1.2 volts, they are tired, but because the Q draws so little high-voltage current, you can probably operate a bit lower, perhaps 1.1 volt(?)
For the high voltage, it should work acceptably down to 24 volts - at least 24v to 36v is the range I designed for.