What is the ruling on shunning a father who committed the sin of adultery in the month of Ramadan, and is such shunning permissible in Sharia?

You are not permitted to be certain of your father's indecency unless you witness it or hear his confession. If you confirm it, advise him and warn him of Allah, and you may involve someone knowledgeable. Otherwise, conceal his sin unless he openly commits it or persists in it. If he repents, you should restore all his rights. If he insists, then you must advise him, accompany him with kindness, pray for him, and obey him. It is forbidden to harm him or be undutiful to him. As for your disdain for him in your heart, there is no blame upon you if it is due to his sin, for disliking people who commit sins is a fundamental principle in religion, as Allah Almighty says: "You will not find a people who believe in Allah and the Last Day loving those who oppose Allah and His Messenger, even if they were their fathers or their sons or their brothers or their kindred." As for avoiding him, if what is meant is not dealing with him as before, it is permissible and may even be obligatory if it deters him from his sin without disobedience. However, if what is meant is abandonment, then it is not permissible.