Assalaamu alaykum. I suffer from incontinence which lasts up to 30-45 mins after urinating. After urinating, I place an absorbent and remove it once the leakage stops, after which I proceed to wash off the impurity from my privates. The problem is that as soon as my private part comes into contact with water, the leakage begins again. What should I do? Should I pray without bothering to wash off the impurity?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

What you mentioned may only be Waswaas (obsessive doubts). If this is the case, then you should not pay any attention to it. However, if what you mentioned is true, then Istijmaar, which is using stones and the like, such as toilet paper, in cleaning the private parts is enough instead of using water.

Hence, when the leakage of urine stops, you can purify your private parts by using tissue paper, so you wipe your private parts three times. If cleanliness is not achieved with three times, then you may wipe again until purity is achieved. This is sufficient to achieve purity, and you are not obliged to use water.

Allah knows best.