Does impurity transfer if it is dry and touched by a very slight amount of sweat, and does it then contaminate objects or mosque carpets that it touches, and what is the ruling on shaking hands with others afterward?

Obsessive doubts have reached a great extent within you. The solution is to ignore these whispers and not pay attention to them. Know that the opinion that impurity does not transfer if one of the two objects is wet is the view of some scholars. There is no harm for you in acting upon this opinion as long as you are afflicted with obsessive doubts. Therefore, do not judge your hand or anything else as impure, live your life normally, and resist these obsessive whispers by ignoring them.