I found out the other day my Lag has a similar problem with the indicator.
In my case, it's not permanently on, but with the engine running, switching on the right indicator, everything is normal…
…until I hit the brakes! Then, the audible feedback (clicking) disappears, and the indicator lamp flashes rapidly.
It's only the right indicator, and it's only when I hit the brakes - And it's only when the engine is running.
If I let go of the brakes, it's back to normal, but due to the nature of automatic transmission,
I obviously use the brakes a lot more when turning in intersections, making the problem even more apparent.
Both in your and my own cases, I suspect relay or some oddball wiring fault, hoping it's not a case of witchcraft on behalf of the