Okay, I was able to fix it. I'm not 100% sure of the order of the steps, because I had been trying various things and failing before hitting on what worked, but I think this is it:
I cleared the Cache (which I had already done several times before) by going to Me>Preferences>Advanced and clicking the Clear Cache button.
I went to Me>Preferences>Graphics and changed the Quality and Speed setting from High to Mid, thinking that this might force a redraw.
I right-clicked on my avatar, and selected My Appearance.
It was either immediately after step 2 or step 3 (can't remember which) that my avatar went totally nude. For the first time, it had a full body again.
In the Wearing tab of the My Appearance window, I clicked wrench icon to the right of Now wearing...
I clicked the wrench icon to the right of the socks that appeared in the list.
I clicked the Texture button, and selected a texture from Library>Textures>Fabric.
Once I hit OK, the whole outfit I had worn before redrew.
I went back and set the Graphics setting back to High.So far, all is well...