I ended up replacing the cable. I bought one from welovemacs for a little bit more than ifixit because I am dumb. But it came overnight and 24 hours later my sound is fixed. The ifixit guides are outstanding. One word of caution, according to welovemacs there are three variants of the same cable, depending on the year model of the MacBook Air. Double check what the site has to say about the right one (my Air is one of the first productions off the line, note the "DATE 0208" on the sticker, so it was simple to figure mine out).
It sounds like the geniuses want to replace the sound board too, but everyone's experience in this forum indicates that it's just the cable. So spend the 40 or 50 bucks, grab some jeweler's screwdrivers, and go to town.
On a side note, I'm totally impressed with the innards of the MBA. I've owned all flavors of Mac laptops, and I completely expected this one to be untouchable inside, owing to it's micro size and the presumed necessity for all the electronics to be one monolithic chunk. Instead I think it's one of the more hackable Mac laptops ever. You can actually identify the speaker wires coming from the sound card. It looks trivial to replace the drive, battery, etc. Also, if you do tighten the screws on the hinges (you should, they wiggle loose quickly) apply some locktite while you're in there.