Does the emission of semen after performing ablution due to cold feet necessitate a full ritual bath (ghusl), and what is the solution to avoid this in order to perform prayers?

It is advisable to consult a trusted physician for treatment. What exits from you resembling manī (seminal fluid) due to cold does not necessitate ghusl (definition) (ritual bath) according to the majority of scholars. Rather, it only requires washing it off and performing wuḍūʾ (ablution). Ibn Qudamah said: "If something resembling manī exits due to illness or cold, not due to desire, then ghusl is not required for it." This is the view of Abu Hanifa and Malik. Al-Shafi'i, however, held the view that ghusl is obligatory. The majority of scholars supported their view by stating that the Prophet, peace and blessings be upon him, described the manī that necessitates ghusl as "white and thick," and he said to Ali, "If water gushes forth, then perform ghusl," and faḍkh (gushing forth) means exiting with force. The other mentioned hadiths are either abrogated or do not apply to this specific situation.