Hard to diagnose an appliance remotely but my guess: Sounds like a logic board problem. I hope it is. You have automatic electronic ignition. It's supposed to keep clicking until it detects the flame is burning. It's not the electrodes but an adjacent temperature sensor that detects the heat of the flame and stops the clicking. That sensor should be failsafe, IE, if the sensor fails or the ignitor fails or any part of it fails, it should keep on clicking.
So the first part of your problem is mundane: the sensor failed, it keeps clicking. But the second part is worrying: If the sensor failed, it should not stop clicking just because another burner is turned on. The problem must be something other than a simple failed wire or sensor.
If you don't know how to diagnose a logic board problem you should have someone who does, look at it. Meanwhile you should be extra cautious making sure that if a burner is switched on, it is in fact burning.