I checked with Appliance Parts Pros.com and found out that the likely culprit was a broken Ice Dispenser Actuator Arm. I watched a video on the same site and figured I could change that part.
I wanted to confirm it was that part so I took the front section apart. The Ice Dispenser Actuator Arm was in fact broken; the little tab that strikes the switch had broken off. While removing that part, when I unscrewed the right side's mounting screw, that section broke off (brittle plastic).
I then ordered the Dispenser Module also. (About $70 for both parts.)
I ordered them on a Wednesday evening and they arrived on Friday afternoon (That's Service!!)
I paid for normal shipping and the did come from within the State, but still~!!!
I managed to remove all the parts (including switches, spring, power panel, etc.) from the Dispenser Module, I installed the Ice Dispenser Actuator Arm; both were fairly easy.
When I tried to re-mount the completed Module into the space, I found it absolutely impossible to get the short hard inflexible water hose back into the hole it came from. I tried soaking it in hot water to make it more flexible, used a shaved small cork stuck into it with a pin and string attached to "pull it down in place". Nothing worked!
I disconnected the two wires to the water switch, folded this crappy hose back onto itself and taped it in that position.
I now have great ice dispensing!!! If I want water, I can walk the 5 long steps to the sink.
Besides, I put a bypass into the filter holder years ago because our water is snow melt and better than bottled.