Will hot-then-cold give me a cold?
Short answer: no — as long as you don't walk out into the aircon still wet.
We're not going to argue with 中医 about wind and cold; you can't win that room and you don't need to. What actually makes people feel off isn't the plunge — it's walking into an air-conditioned room with wet hair and a cold body.
That's exactly what the last ten minutes of every session are for: you rinse, you dry off properly, you sit on the benches until your body settles and your heart rate comes down, and then you leave. Nobody's rushing you out. The whole routine is built so that leaving cold and damp simply doesn't happen.