Thanks to all the responses so far.

To the person that said to put a finished backer to put the crown
molding flush with the doors rather than the face frames: Thanks! I
went to a cabinet store today and saw that they had them flush to the
frames, and it looks much worse that
way. I ordered backing molding today and I'm sure it will look much
better.

To the person that said to drop the cabinets down and leave a space
above: I already have the cabinets. But I deliberately chose to have
them go all the way up because I like that look better

To the people that think I should just put a level, flat drywall
ceiling over the existing ceiling: the current ceiling isn't ugly,
just unlevel, and to me the idea of putting another over it seems like
a lot of work. This is especially true because I am doing almost all
of this alone and it is hard to get a helper. Renting a drywall
lifter? Hauling it up, screwing and taping it all on the ceiling by
myself? If there is another solution I am interested in it. But I
appreciate your opinions.

I talked to another cabinet guy today and he suggested that I simply
run some scribe molding, which is easily bendable, along the top of
the crown to the ceiling. The crown has about 1/2" that the scribe
could run up and down if it turns out that the gaps are too much to
caulk over. I'm a bit surprised that no one touched this idea--is
there a problem with this? It doesn't seem to me that it would look
bad, or would even be noticeable to most people.

I think the best strategy now is to hang the cabinets so the crown is
flush with the ceiling at the low point. If the gap is small enough to
caulk, I'll do that. If the gap is too big, I'll run scribe along it
and bend it to the ceiling. Please let me know if there is something
wrong with this solution.

Also, a follow-up question: does anyone have any experience with using
a high-tack construction adhesive like Lok-Tite to apply molding? It
seems much easier than pre-drilling and nailing, and there are no nail
holes to fill. I don't have a brad nailer. I might buy a
compressor/brad kit if it really seems necessary but I thought I'd ask
about this first. The video demo at the store shows them putting it on
a wall and immediately hanging a brick on the stuff!

Thanks again for the helpful responses and I look forward to more of
them.