If your mother is a Christian then you have to call her to Islam with wisdom and beautiful preaching. Tell her about the beauties of Islam and its characteristics and beliefs. Explain to her the falseness of the Christian belief in the divinity of Jesus (peace be upon him). You can also refer to the questions in the Da’wah to Non-Muslims section of this website, so that you can convince her, as well as reading some books which speak of the falseness of Christian doctrines.
But if your mother is a Muslim, then you have to call her to give up sin and to repent from it, especially drinking alcohol. Tell her the evidence from the Qur’aan and Sunnah which proves that it is haraam, and that when a command comes from Allaah or His Messenger (peace and blessings of Allaah be upon him), the believer has to hear and obey, and hasten to follow the command even if that is contrary to his own whims and desires.
Make her fear Allaah and His punishment in this world and in the Hereafter.
You have to be gentle and kind to her, tell her that you are eager that she should be guided and saved from Hell.
Perhaps this will be the means of guiding her to the truth.
With regard to the cooking utensils with which haraam foods are cooked, if there are others available then it is preferable for you to eat from those. But if no others are available, then you can wash them with water and then eat from them.
It was narrated that Abu Tha’labah al-Khushani (may Allaah be pleased with him) said: “I said, ‘O Messenger of Allaah, I live in the land of the People of the Book. Please advise me with regard to their vessels.’ He said, ‘Do not eat from them, unless you cannot find anything else, in which case wash them and eat from them.’” (Agreed upon. Buloogh al-Maraam, p. 23, hadeeth no. 24).
With regard to your thinking of leaving home or leaving your country, that may be better if you know that your family will not respond to your words and if staying with them will damage your religious commitment in such a way that you cannot do what Allaah commands.
But if you know or you think it most likely that after you call them to Allaah they will respond, then you must stay with them.
Imam al-Bukhaari included a chapter in his Saheeh entitled: Baab al-Inbisaat ila’l-Naas (Chapter on being friendly towards people). Ibn Mas’ood said: “Mix with people but beware of damaging your religious commitment.”
(Saheeh al-Bukhaari, 5/2270).
But if you know from your experience with your family that this will be futile, then leave them and go and live in a place where you can practice the rituals of Islam.
We ask Allaah to make you steadfast in your religious commitment and to keep you safe. We ask Him to guide your mother and all your family to His religion so that the religion of Allaah will be the dearest thing to you all.
And Allaah knows best.