After clearing out the overflow pipe, I still had a problem because water has started dripping in places not directly over the drain pan.
I "extended" the coverage with sheet metal, but the water is still dripping from somewhere that I can not yet see. I looked inside the condenser housing (which is directly above the pan) and saw that water is standing there.
Is there supposed to be a drain hole somewhere on the bottom of this sheet metal housing? (Water backing up in there might be getting deep enough to flow into the much larger distribution box where the ducts connect.)