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I have the same issue, it's actually drained the battery once. Mine was that the relay was so worn out that it was able to connect with no power energizing the relay. You said you've tried replacing the relay already, but have you tried smacking the old one? That would get mine to stop for a few minutes. If that makes it stop clicking for a few seconds, you probably got a bad relay from Amazon. If that does nothing, then there is probably a bad connection on your turn stalk, since the hazards and the turn signals are usually on different fuses.
With the turn signal fuse out, try turning on your emergency flashers. If the relay energizes and stops once you turn them off, then it is most likely the contacts on the stalk. You'll most likely need to remove the steering wheel and plastic trim on your column to get to it so you can see where the short is happening.
If the hazards didn't turn on, but do turn on when you reinstall the fuse, then check that switch as well.
It ended up being the whole blink switch/handle. I was able to replace that pretty easy and the flasher relay clicking stop. Glad it wasn't something more severe.
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