Turns out I get the same behavior if I pick a part from one of the included libraries.
If I select the part in the "Project Explorer", then I can move it. But in the schematic screen, I cannot select and move the part.
Please tell me what I've done to make this happen. I have been playing with the screen colors, but I can't find anything else that I have changed that might cause this.
I didn't have the "select components" icon selected at the top. Sorry for the false alarm.