I still have the demo installed and was able to patch just fine (about to run my game and see if the game works).
When I first opened my launcher, I didn't see any game updates and got the message about not able to connect to the EA Launcher so I closed it. I then opened my Origin and while that was open, re-opened the launcher. I was then able to patch. Of course, before I patched I renamed my Sims folders to protect my content following crinrict's advice.
EDIT: Forgot to mention that when I installed the demo, I had renamed my main Sims folder and the demo made its own set of folders. I later renamed these folders so that it'll be separate from my main game, i.e. "Sims 3.pets"