FOLLOW UP: The service guy from the warranty company just left. He took the bottom panel off at the BACK of the refrigerator, went in and removed a rubberish sort of funnel, or something that was part of the funnel. Being rubberish, in the heat the rubber is all sticky and this gizmo was stuck shut where the water is supposed to drain through. He simply removed that part, and said they send it in to Maytag for analysis. It took him all of about 7 seconds to diagnose this once he saw all the symptoms, so I gather this is a common problem. I hope it is ok to run it with that item out.
He said the actual drain is frozen solid, so we have to turn it off and open the freezer door for 24 hours. I was hoping he could have taken care of it without that, but that is what we will do.