The cure was to replace the "Flasher Unit".
The apparent cause was normal thermal wear, that limits the part life to several years. This was probably exacerbated by the high air temperatures that characterize South Carolina. The probable mode, is slowly rising resistance in the coil, causing a rise in idle temperature over the years, until combined with high air temperature it is enough to keep the part oscillating continuously. The indicator switch keeps the actual indicator lights from blinking. I suspect the reason for the "always on" feature is to prevent delay in starting to blink when needed in cold climes.
Without the service manual, it took ~60min to learn how to remove the lower cowling, and another hour to learn how to free the old flasher unit with a standard clip tool. In hindsight, the service manual would likely have made clear how to replace the flasher unit without removing the cowling, and directed use of a clip tool. The cure should then have taken about 10min.