The people at Clinica Valparaiso are very nice and helpful. The doctor said I have a virus, which is sort of obvious since this started with having a cold. The sweating and chills are "toxins from the virus"; I don't even have a fever. The dizziness/balance issues are because both my ears are seriously obstructed by earwax. (I'm not sure of the mechanism on that, but it seems intuitively possible and the Internet agrees.) She prescribed a particular over-the-counter anti-cold thinger, and next week I go see an ENT doctor about the ears.
In the meantime, I seem to be needing a 1-hour nap for every 3-5 hours of activity, which is lame, although I do feel less awful than yesterday. Hopefully I can go in on Monday to run tests with the kids.