If you are exerting yourself during your swim lessons, then your body temperature is elevated when you get out of the pool and you'll be sweating even though you don't feel it while you're in the pool. You need to lower your body temperature before you put your clothes on. There are two ways I've used to cool off before getting dressed.
If the pool temperature is typical (around 80 degrees F), i.e. not too warm, just stay in the pool for a few minutes to cool off. Your normal skin temperature is usually above 90 degrees so full immersion in 80 degree water will cool you off. The longer you can stay immersed in lower temperature water, the cooler your body will be when you get out of the pool.
Take a cool or cold shower. Most people like the feeling of a hot shower but if you're already overheated, this just makes the problem worse. You'll be sweating worse after a hot shower. Take as cold a shower as you can stand and spend extra time with the cool water directed at your head where your body's thermal regulating system is located. A cool head is a cool body. Again, the longer you can do this, the lower your body temperature will be when you get dressed.
Ideally you should do both of these, but obviously time is the constraint here. As a final measure, I sometimes leave my shirt unbuttoned and untucked until I cool off a bit and use my towel to blot up the sweat off my body for a few minutes.