I have never heard of mold sneaking in. Do you happen to live in some tropical climate, or are you just worrying?
If you don't already have one, get some kind of reed case which holds the reeds gently but firmly onto a flat surface such as plate glass. I can't stress strongly enough how important to reed life it is to keep the bottom face flat.
The reason to swab out instruments (notice that metal ones such as saxophones and trombones get swabbed too!) is to keep them clean, avoid soaking the pads themselves, and so on. [use cigarette paper to clean and dry the pads themselves] The wood doesn't care if there's a little moisture there, tho' you should swab the interior now and then with a small amount of bore oil