Interesting. On our '06 Dynasty, the turn signal only "clicks" if the engine brake is not enabled, or the brake pedal is not depressed. However, even when it is not "clicking", the selected turn arrow indicator on the dash is still blinking. The fact is, I haven't checked out side to verify that the blinkers are actually blinking. But this has been the way it has functioned since it was new.
On a side note, I HAVE A THEORY - Now, I'm not talking about our Monaco's, but rather our conventional vehicles. I haven't researched this, at all, but I don't think that most vehicles even have the old conventional blinker-can's that we all grew up with. It is my belief that the blinking function has been electronically produced for several years, and that the clicking sound that we all probably need to make us feel like everything is working properly is produced by an audio circuit and a speaker. I haven't had a flasher failure in any of our cars in over 40 years, and I can't believe that those mechanical thermally operated cans are THAT reliable. Or, there may be a relay being driven that produces the "clicking", but I believe that the actual blinking function is electronically produced. In other words, in many vehicles I wonder if there even is a good old fashioned "flasher".
On our coach, I developed several schematics that work for me. One was the brake light circuit, and it includes the flasher. I just looked at that drawing, and the flasher is schematically located above where the circuit leaves the front run box and before it arrives at connector M10. If it helps, on ours, the wire leaving the front run box is black to the flasher. Leaving the flasher, it is black as well, and splits to two purple wires that go to M10 pins 1 and 11. So maybe you can trace it from one end or the other. But I agree that it is probably under the dash, or behind the panel beside your left knee in the drivers seat. Good luck!
Jim & Angie, Boise, ID
2006 Monaco Dynasty Countess III
2017 GMC Acadia Denali