Is the prayer of one who performs tayammum due to obsessive-compulsive cleanliness, which prevents him from performing istinja’ and istijmar for fear of splashing, valid?

You must beware of waswasah (whispering doubts). If you have cleansed yourself with water, you are purified. Do not pay attention to the whispers of Satan. If you fear using water, it is sufficient to remove the impurity with tissue or the like, provided it has not spread beyond its usual place. And if any splash of water spritzes, it is excused. It is not permissible for you to pray with tayammum (definition) (dry ablution) as long as you are able to use water for wudu (ablution), and your prayer with tayammum in this state is not valid. You must make up all the prayers during which you were able to use water and had it available.