Is it possible to be too self-aware?

yes
self doubt will plague and infect your mind at every waking moment

Only if you don't do anything with your self awareness, I feel like.

If you're super aware of your flaws but you don't do anything about them, you'll just feel increasingly shitty about yourself. That's about where I am right now, actually.

Yes, but if you accept it and work with it you can turn it into an advanced state of mindfulness