It was clear that the work was impossible, but the committee wouldn't budge on the fact that I had to finish it. I was working around the clock and then laying in bed for a few hours worrying before working some more, and I absolutely couldn't take it anymore.
I decided the only solution was to drop out, which meant the postdoc would rescind my offer, and I'd have to break the new lease, cancel the move, etc. I went into the admin office to let them know I was simply not coming into the lab anymore and they could do whatever they wanted to do about it.
Suddenly the "must do" list evaporated and they said I could just write up my work and graduate. So I did, and I got the phd and went off to the postdoc and it all turned out fine-- it turns out all I needed to do was decide to throw my life away and really mean it, in order to call their bluff.
I think the experience gave me a better sense of when I'm approaching the burnout zone, so that I can better avoid it. I've also never experienced anything even close to this in a work environment -- if my job was like that, I'd quit in a hot second and go work somewhere not-awful. It was only the fact that they were holding the degree hostage that caused me to feel forced to overwork myself into an unhealthy state.