How can one get rid of the obsessive thoughts related to purity, such as water splashes during istibra' and washing hands, repeating wudu', urine drops, and washing clothes?

The cure for obsessive thoughts (waswas) is to ignore them. Whatever the devil whispers, whether it's about impurity or anything else, turn away from the whisper and do not pay attention to it until you attain absolute certainty that the impurity has occurred. This might be difficult at first, but with determination, seeking Allah's help, striving against oneself, and a strong desire for recovery, healing will be achieved, Allah willing. Giving in to obsessive thoughts is contrary to the Sharia (definition), which has placed no hardship upon us.