LG G3 :: Google Apps Not Working - Can't Delete Google Account

I have a really weird problem that randomly started and just keeps getting weirder and more annoying. I was watching YouTube via the app on a very unreliable WiFi connection, and received a number of "playback error, tap to retry" messages (pretty normal on this particular network) so I didn't think much of it. When I got home, however, the messages and errors continued, so I cleared YouTube's cache, which did nothing. I then cleared data, it asked me to sign into Google again, and it did, but the errors continued. I then noticed in my notifications that I had some app updates available from the play store, and that's when the problem got stupid.

Play store hung on the spinny loading circle, so I cleared its cache and data and then it hung on the first update. Soooo, I cleared cache, then data, on Google play services. Now, not a single damn Google app does anything. Google play music shows my two google accounts to set up with, clicking either of them does nothing. YouTube plays videos linked from other apps, but can't search or browse, just has a the red nav bar and blank white space with the spinny. Play store doesn't work. Inbox doesn't refresh, you get the idea. And somehow, Facebook started not connecting at the same time as everything else. Not sure if its related or not, but it sure is frustrating. Chrome works fine. Other apps that connect to the internet work fine. Just google apps and Facebook.