Solution: Replace the Flasher Module
I couldn't find a solution in the forums, so now there is. I consistently found that the left turn signals didn't come on every other attempt, even in back-to-back attempts, which was puzzling. When it wasn't working the hazards didn't turn them on either. I checked all the bulbs and connections, and got a lead that the flasher module failed frequently on the left side only, and spent $80 on a new module. That fixed the problem.
I couldn't find a solution in the forums, so now there is. I consistently found that the left turn signals didn't come on every other attempt, even in back-to-back attempts, which was puzzling. When it wasn't working the hazards didn't turn them on either. I checked all the bulbs and connections, and got a lead that the flasher module failed frequently on the left side only, and spent $80 on a new module. That fixed the problem.