Left of the center dash AC vent is the Hazard Warning Flasher. It costs $35. That is your culprit. Unfortunately, you need to take out parts of the dash to replace it.
If you search the Forums, you will see instructions on how to take apart enough of the dash to get to it. My buddy who owns a repair shop did it for me. He said that he got to it by removing the stereo and reaching up, and did not have to remove to dash pad.
Meanwhile, if the turn signals stop working, you can probably get them to work again if you either 1) bang on the dash above the Hazard Switch, or 2) push the Hazard switch in and out a bunch of times.
I guess you can tell that I'm right on top of GM engineering issues!
Good luck!!