What is the ruling on touching dry, impure dust that has spread throughout the school, given the presence of sweat on the hand and the difficulty of purification?

It is clear that the questioner is afflicted with waswasah (obsessive whispering), so it is advised to ignore it and pay no attention to it, and to act according to the easiest opinions in matters of Shari'ah. Many scholars believe that the earth and what is connected to it become pure through dryness. So, if the impurity dries up by sun or wind and no trace of it remains, then the dust is deemed pure. This is if the dust was originally impure; otherwise, the default is that impurity does not transfer when in doubt.