Sometimes a magic wand can be handy. Like when you need to use a computer that isn't yours and you don't want to leave any data behind. In Short Circuits: All operating systems have a couple of useful utilities that can show where a problem is when you can't reach a website or any other device with an IP address.
It turns out the issue appeared to be my cable box, not the TV. I set the audio output on the box to PCM only instead of Dolby, and suddenly the tv kept sync (I still had to set the sync settings on it, but prior to that, I'd do so, and after 10 minutes it would be out of sync).