What is required when madhiy is emitted
It seems that this wetness that you feel is madhiy and not maniy, the emission of which means that you must do ghusl.
Based on this, you do not have to do ghusl when it is emitted, you only have to do wudoo, because the emission of madhiy invalidates wudoo’, as the Prophet (peace and blessings of Allaah be upon him) said concerning madhiy: “Wudoo’ is required for it.” Narrated by al-Bukhaari and Muslim.
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (1/168): The scholars are unanimously agreed that the emission of madhiy invalidates wudoo’
It must be noted that madhiy is naajis (impure), and any part of the body that it touches must be washed. As for clothing, Islam is lenient about purifying it so as to ward off hardship. It is sufficient for you to take a handful of water and sprinkle it on the place that has been touched by madhiy, because when the Messenger (peace and blessings of Allaah be upon him) was asked about how to cleanse a garment of madhiy, he said: “It is sufficient for you to take a handful of water and sprinkle your garment where you think the madhiy touched it.” Narrated by al-Tirmidhi (115); classed as hasan by al-Albaani in Saheeh al-Tirmidhi.
We have explained the difference between madhiy and maniy and the rulings on them in the answer to question no. 2458, to which you may refer for more information.
And Allaah knows best.