How does one rid himself of the obsessive doubt regarding the impurity of hands or clothes, and what is the ruling on such obsessive doubt in Islam?

The default state of things is purity until their impurity is ascertained. They are not ruled impure merely based on doubt. This is part of the ease of Islamic law. Indulging in whispers (waswas) in this regard is an act of obedience to Satan. So, whenever Satan whispers that something is impure, disregard that whisper and build upon the default principle, which is purity.