What is the ruling on recurring doubt about water drops coming out of the male organ after istinja' (ablution after defecation or urination)? And is it obligatory to inspect in this case, or is quick disregard and drying the male organ with a tissue after istinja' sufficient, especially if the doubt is persistent?

The cure for obsessive whispers (waswas) is to ignore them and turn away from them. So continue to disregard the whispers of Satan about something coming out of you. Act upon the original presumption, which is that nothing has come out. Do not judge that something has come out unless you have certain, definitive assurance that you can swear by. Do not inspect your private parts or your clothes. Rather, act upon your certainty that nothing has come out, and do not pay attention to doubt.