If you can't get it to be problem free without the engine running (which eliminated the engine as a cause), you have only two choices. Replace the crystals, and or module if you have one, or send the primary radio components off to the manufacturer for a good check up. I would go the check up route in any event. See if you can borrow someone elses Tx and Rx for a system check. If that works well, you know the problem is with your Tx or Rx. If it doesn't then you have a battery or switch problem.

20 paces (60 feet) is a terrible range test with the antenna collapsed or removed. I wouldn't consider flying the glow plane either.