Car won't turn off
 

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As you can see, my car won't turn off no matter how long I press the button. Also, I noticed that the headlights turn on and won't go off while this is happening, and that the car won't start unless I put the key fob in the slot. Any thoughts?

Check for water in/near/around the passenger floor/kickpanel area. Your BCM (Body Control Module) controls those functions and if it gets wet can cause a ton of problems. Typically, if you press the button 3x or press AND hold the button should force the car to shut down. Again, depends on the BCM condition.

Yeah I recently noticed that there was some water coming out of the carpet on the right rear seat's floor, but not the passenger side in the coupe (which I find weird as almost everyone I see has a leak either on the driver or passenger side instead of the rear). I am waiting on receiving a new drain plug before taking out the seats and intensely drying everything as it seems to be a sunroof leak. Do you think the problem would arise after I dry it or would the BCM need to be replaced?

More than likely the water is saturating the front passenger area and "overflowing" to the rear. Rear drain failures are highly rare, but not impossible. Also check the wiring harness firewall grommet and your door seals (especially near the mirror).

Some have had luck with the BCM returning to normal after letting it thoroughly dry out for a period of time while others have had to replace. There is no way to predict.

It is strongly recommended you disconnect the battery before working on any electronic components.