How can a married woman, who is married to a Christian alcoholic and has two children, convert to Islam while desiring not to leave her husband due to his reliance on her, and at the same time harboring feelings for a married Muslim friend who is much younger than her?

The questioner must embrace Islam and present it to her husband. If he converts, then that is good. If he does not, she will enter into her waiting period ('iddah (definition)). If it expires without him embracing Islam, then separation between them becomes obligatory. As for her relationship with the other man, it is an illicit relationship, and she must sever it.