The AC guy showed up and confirmed that it was indeed a drainage issue. He fixed it in a lot less time than I thought it would take - 30 minutes, maybe 45 minutes total! I didn't pay close attention but he just needed a hose and bucket, and what I assume was a pump he brought along since I heard a mechanical whirring in the background.
However the carpet still needs to be addressed. It is stained and discolored along the edge. Right now I have all the furniture stacked up in one corner of the dining room, and will probably leave it there while I decide what to do about carpet replacement.