I'm actually elbow deep in working on my indicators, have installed some LED clusters with integrated brake lights. Unfortunately, there are few spots where you can expect something easy to fix if you have a short. You can open the indicator housings themselves and verify that the cables are still attached and not shorting in there, but the likelihood is that something has rubbed through and is now connecting to ground. And that can be... well... in a lot of places.
Anyway, here's my suggestion on how to get started, for what it's worth. I make no claims on being an expert.
Also, I have a Touring so that might influence this vs yours.
If I were you, I'd start with checking the horn. That one is right out there up front on the bike, and the most exposed. Verify that the cables are attached and not shorting.
First, though, disconnect the positive lead on the bike's battery.
Disconnect the horn leads and measure resistance with a multimeter between that connector and ground (put one of the probes on the multimeter against some grounded metal part on the bike). One connector should have near-zero resistance, the other should have no connection at all when the bike is off (the positive one).
You can then try doing the same in each indicator housing (open them up, pull out the innards gently and disconnect the cables and measure the cables individually to ground.
If any of the four positive wires show connectivity to ground, you have a place to start. You can at least narrow it down to the left/right side of the bike (front+rear indicator form one circuit).
Though the fact that they work when the engine is off might mean they only short out when the bike is vibrating, that should be a fun one to diagnose, good luck.