Hey, Methuselah! Before anything, I want to compare with you on what versions the Blizzard Application and WoW are currently on.
For the Blizzard Application, click on the Blizzard Logo at the top-left corner, and go to Settings. At the bottom-left corner, you should see the version 1.23.2.12210.
For World of WarCraft, the version should be on the right of the play button. It should show version 8.3.7.35435 and for Classic, 1.13.5.35395.
One of the most common issues with the Blizzard Applications not pushing the update is because its not updated itself. A clean reinstall should do the trick.