Here is an update. Since we arrived home on September 16th, I have been patiently troubleshooting this problem. I'm not retired for 64 more days, and there is not much light left when I get home from work these days. So, far, I can turn on the lights, without the towing cord, and they stay on; at least they have for up to 1 hour, and coming home the were going off every few minutes. When I plug the towing power cord in to the coach only, they also stay on indefinitely. When I plug the cord in to the toad, the lights went off within a few minutes. I changed both connectors on the cord, and everything is the same as mentioned.
My next step is to totally trace the wires for the female connector on the toad from the grill to the auxiliary lights we installed in the light assembly in the rear. I will report what I find.
Stan