Hopefully sound cards are inexpensive enough that I'll just buy one, unless there's some really convenient way of transferring possession of it.

I'm kind of guessing here, but the symptoms don't go away when I swap the speakers for other speakers or headphones, and I get the same symptoms from the front audio jack as from the rear one. (The symptoms are not a lack of sound, but that the sound is, in technical terms, fucked up.)