Doesn't sound good, but is there a way to reset the touch screen? That would be the first thing I'd try, and the procedure (if it's possible) will depend on the phone manufacturer.

Next you might try a factory reset (always a last resort in my book, it's such a hassle getting everything back where it was) - on my Motorola Droid Mini it's under Settings>Personal>Backup & Reset. But you'll want to back it up first, obviously, which you'll back it up to Google...

Is it even possible to log out of Google on an Android phone? Android is a Google product. My phone doesn't let me log out of Google. I can log out on the browser (which I use Firefox for whatever good it does), but not on the main phone account. Logging into Google is the first thing you do when you set up a new Android phone.