... but I'm unstable and generally incompetent.
Yeah, so is basically everyone else. I really wouldn't worry about it, except from the perspective of understanding what about yourself you can seek to improve, which is tangential at best to the job thing (makes good interview fodder, that's about it). They'll hire you for what you're good at, which is already a lot of things, and seems to turn out to be most things you throw yourself into. There aren't many options if the job opening demands an emotionally stable, omnicapable supergenius. Usually they get to pick one, maybe two, of those qualities, if they're lucky.
If you need to figure out what makes you a useful tool for an organization, it's less about what you can't do and more about what you
can offer. As long as you're not so far below average that it outweighs everything else, you're fine - and if you can graduate from Berkeley, you meet that goalpost no matter what the particulars were. Just handing in homework consistently* on time demonstrates that level of achievement.
*Also counts if you didn't, but worked with your professors to arrange a satisfactory system. Point is, if you were able to get a passing grade, you put in the effort to ensure you worked within the system to a reasonable set of expectations, which is basically what they want out of you.
EDIT: As for becoming less hopeful about the future - don't worry so much about it as a destination to be reached perhaps? I don't know if that's a thing that's really relevant here, but trying to think less of the future when it comes to my plans for life, and more of an extension of the present into more of itself, has helped me when I start feeling nervous about having no idea what I'm going to do with my life. Redefining where I'm going to where I'm going to end up is sometimes helpful, as long as it doesn't become a "nothing really matters" mantra. Many things matter, but what I'm supposed to be trying to do isn't fixed.