Submitted my application to YC the day (after) my idea demoed at YC

Will you fund multiple startups working on the same idea?

Yes. If you fund as many companies as we do it's unavoidable you'll end up with some overlap. Even if you tried not to accept competing companies, you'd still get overlap because startups' ideas morph so much. The way we deal with it is that when two startups are working on related stuff, we don't talk to one about what the other's doing.

In practice it has not turned out to be a problem, because most big markets have room for several slightly different solutions, and it's unlikely that two startups would do precisely the same thing.

To be perfectly honest if I were in Y Combinator's shoes I'd be more concerned about you seemingly giving up (from your post: "this is basically a pre-mortem for ShopJoy"). If you believe in your idea and your team, then you'll succeed with or without Y Combinator. The path of a startup is filled with roadblocks and the question is how you handle them. So show that you haven't given up! Forget about this and focus on your product, and by the time you hopefully get an interview you can show progress.

Good luck!