i personally like the plan of b) apologize (etc etc) largely because it's nice, and you're stressed, and the prof might be a little concerned (I once got an email from a prof asking me if everything was ok and if i'd like to hand in a paper late, full credit), and I know that, for me, writing emails and getting responses back is like crack. But good luck on whatever course of action you choose. (Also I frigging hate it when people tell me that i "should've done something", like, yeah sure, just give me your time machine and I'll fix that right off, you pretentious asshole. So heavy sympathy on that front.)
Yeah, I think at this point you're best off talking to your prof, apologizing, explaining the situation, and trying to rest/move on with your life. Who knows - maybe they can give you more time to write the essay? That or they may suggest that it's not worth your time, so don't bother. They may also know the best way to navigate the bureaucracy of withdrawing and stuff. Also, I've met more profs who were concerned about their students' well-being, and about their students finding the right classes and careers, than who were concerned about coursework. (Not to say they don't exist... I've met a few of those, too.) I suspect you'll find more sympathy than shame, if they're a decent prof.
I like the one where you talk to the professor and explain that you're basically dropping out due to depression/stress and that because of this, the essay's not done, but ask if he'd be willing to let you hand it in maybe a week later so you can try to do a good job of it. And basically, the general rule I use is: If someone is expecting a thing, telling them that it won't be available, or when it might be, is almost always appreciated.
You could say that you suspect you are depressed but haven't been able to see a professional yet to get evaluated, which is true.
It mostly works pretty well. Sometimes you get assholes who are just mad at you and say stuff like "you're not depressed, you're just lazy", but, well, they're assholes, they wouldn't necessarily have treated you any better with a formal diagnosis. Also most people who are not idiots are aware that depressed people can have trouble seeing a professional.
Yeah, I was about to suggest asking for an extension. Especially this far in advance, most professors I've had are more than willing to give it. I've cited anxiety attacks and such for not being able to do things, and no one has ever asked for proof or anything, so I think it'd be safe to just say you're depressed. Especially since you're going to be taking time off school. That's enough proof for most people. And especially if the prof has been working with you, they should understand and be willing to give you more time (if that's in their power).
Sorry to hear that things didn't go as you'd hoped! *hugs if desired* Keep your chin up, it'll all come out in the wash. :) Good on you for giving it a go and trying to figure it out. That's worth a lot.