I went through the steps you outlined, and when I chose the perspective filter, it gave me the four handles in the corners, which was encouraging. Previously, I was getting a blue square around my sign JPEG, and the edges of the square did not line up with the edges of my image, and there were no handles on the square. So seeing a selection box that matched my image and seeing the selection handles gave me hope.
However, when I move the handles of the box, the image does not move with them. And when I click Apply, no perspective changes have been made to my image.
The image I'm working on is for an ad for a new restaurant coming to our town. Even though I couldn't get the perspective tool to work, I did manage to mock up a sign for them. (I think I converted the layer to curves or something. That allowed me to line up the corners of the new sign over the old sign, but it didn't adjust the perspective at all.) They approved the ad today, so that's good.
I guess I will just hope that next time I need the perspective tool, I'll be working on a fresh image and won't run into any problems!
I appreciate everyone's help. The problem isn't solved, but I appreciate the time you spent and the ideas you offered.