What is the solution to the problem of continuous pre-seminal fluid discharge (madhy) and the obsessive thoughts resulting from it, which affect the performance of prayers at their prescribed times?

Both manī (seminal fluid) and madhī (prostatic fluid) have distinct characteristics. If you are confused and doubt which of the two it is, you may choose to consider it manī and perform ghusl (ritual bath), or madhī, in which case you perform wudu' (ablution) and wash whatever parts of your body or clothes it touched. This is according to the explicit texts of the Shāfiʿī jurists.

Expel these intrusive thoughts from yourself. What you experience regarding the discharge of madhī is common among young people, and it was also present among some of the Companions.

On the authority of Ali (may Allah be pleased with him), he said: "I was a man who discharged a lot of madhī, so I felt shy to ask the Messenger of Allah (peace and blessings be upon him). So I asked Al-Miqdād ibn Al-Aswad to ask him, and he said: 'For it, one performs wudu'.'" (Agreed upon). In another narration by Muslim: "He washes his private part and performs wudu'."

It is not permissible to delay the prayer beyond its time because of this discharge. Rather, you are obligated to purify yourself from it, perform wudu', and pray before its time expires. If you find water, even if by purchasing it, you are obligated to perform wudu'. If you do not find it, you are obligated to perform tayammum (dry ablution) and pray within its time. Do not miss Fajr prayer for this reason, even if ghusl is sometimes required of you.