Internet: Connected, yet not actually connected

This problem is specific to the iMac identified. All other iMacs, MacBooks and other devices continue to work normally on the same network.

If my iMac does a self-imposed shutdown, or if I shutdown or restart, all apps requiring internet act like there is no internet connection. If I go to the Network System Preference Pane, however, everything looks normal (and it is for my other devices). Misc acts like switching wifi on and off, etc. fail to fix anything.

BUT, my app connections eventually come back, if I absolutely do not shut my computer down at any point. It could be a few hours, but most likely a few days or as much as a week. I monitor my status by keeping Speedtest running and try to run it from time to time. If that works, all other apps requiring internet also work again. As long, as the Mac is NOT ever shut down for any reason, the web apps will continue to function normally. Otherwise, same vicious cycle. Inevitably, the iMac itself finds a reason to shut itself down or do a restart, and I'm screwed again.