Hi everyone,
Well, after almost 4 hours of derailleur fiddling, I'm proud to report that I have figured some things out! I found a key piece of advice from the FSA website - they suggested raising the front derailleur 1-2 mm higher than the normal recommendation, to solve the chain drop problem. This seems to work for me - the derailleur now sits 2 dime-widths above the big ring teeth, versus 1. It does slow down the shifting, but the shift is much cleaner (chain not bouncing around like crazy), so I think I can live with the slower downshifting.

In the process of undoing and reattaching the cable many times (where I had zero prior experience), I seem to have created another shifting problem having to do with the indexing. Both my downshifting and upshifting now take 2 pushes. From big to small, the first push moves the chain, but then after the shift I have to push the lever again (same direction) to trim it. From small to big, the first push moves the derailleur closer to the big ring, and then it takes a second push to get the chain up to the big ring.

Could I somehow have messed up the indexing of my STI levers while undoing and reattaching the FD cable?

Well, at least this new problem is less annoying to me than the chain dropping all the time!