The model A ignition HT wire radiates a lot of RF interference. We had a backing camera on a 12 volt Town Car that would light up with the engine off and and go blank with it running. With the car turned off the camera was going and when another model A right next to the non-running one was started, the camera in the non-running one went blank. So I replaced the copper core coil HT lead in the car with the camera that would not run with a resistor (carbon core) one and the camera immediately worked properly with its engine running.
With that experience in hand, I later had the same problem with a 1952 2.5 litre Riley, but this time it was the LED indicators that did the same thing with an electronic flasher unit. First I tried a separate battery for the flashers, then the car battery circuit for the flashers and the separate battery to run the car ignition. Still the indicators flashed with the engine not running and did not when it ran. Shielding the plastic electronic flasher can in a tin can connected to earth did not solve the problem, nor did wrapping it in earthen aluminium foil.
So, rather unwillingly, I put in a carbon core HT lead and immediately the flashers worked with the engine running. The spark was not as strong on my spark gap tester, but still in the OK zone and the Riley still started well. There were about 5500 ohms resistance in the wire and I kept cutting it shorter until at about 3300 ohms the indicators stopped working again with the engine running. So I put in a longer piece of carbon HT wire having about 3700 ohms, everything worked and I left it at that.
So maybe try a piece of carbon core resistor wire for the HT lead and if that makes things work you will have diagnosed that it is RF interference that is upsetting the flashers. What you do from there is your decision. I personally like copper cored HT leads in 6 volt Model As, so have no backing cameras in mine.
My Model A flashers are incandescent bulbs in the front and LED in the rear with bimetallic flasher cans in both cars. I added load resistors at the front in my roadster and hidden extra bulbs behind the dash in my wife's 6 volt Tudor to draw enough current in both to heat up and make the bimetaliic strips work.
RF from the HT leads may not be your problem, but putting in a bit of old carbon core thrown out from modern car is an easy experiment, as long as you check it for continuity with an ohm meter first to make sure it was not discarded because it broke Internally!
SAJ in NZ