The indicator lights in my 2008 Ford Ranger PJ are stuck on

The symptoms here point to a stuck flasher relay. This is the electrical component that not only powers the indicator lights, but makes them flash (and produces the steady clicking sound you hear when the indicators are on). The contacts in these units can become stuck at which point the lights can become stuck on.

Why does it still happen when the ignition is off? Because the flasher relay is powered up even when the ignition is not. That’s so you can have the emergency hazard-warning lights flashing even though the vehicle is locked and the keys in your pocket. A new flasher relay unit should fix it.