Praise be to Allah.
The conviction that you have reached must be very firm and strong, because it is based on research and study, and comes after you have compared the Bible and the Quran. We do not think that that you need to be convinced any further, but the problems that you talk about have to do with the next step, which is really embracing the religion of Islam and starting to practise it by living an Islamic life. We do not see any real obstacles that would prevent you from taking this step. Let us look at each of these problems in turn.
The first problem is: how will your alcoholic husband react to your becoming Muslim? The Islamic ruling is that as soon as you embrace Islam, you start your iddah, which is the waiting period (following dissolution of a marriage).
Maalik (may Allaah have mercy on him) said: According to our opinion, if a woman becomes Muslim when her husband is a kaafir, then he becomes Muslim, then he has more right (to go back) to her as long as she is still in her iddah period, but if her iddah has expired, he has no right to go back to her.
Al-Shaafai said: If one of them (the couple) becomes Muslim before the marriage is consummated, then their separation takes effect instantly when one of them becomes Muslim. If this happens after the marriage has been consummated, then it becomes the matter of waiting until three (menstrual periods) have passed. This is if the woman has regular periods; otherwise the waiting period is three months. If the other partner becomes Muslim before this time has passed, then their marriage contract remains valid.
(Tabyeen al-Haqaaiq Sharh Kanz al-Daqaaiq, part 2, Baab Nikaah al-Kaafir).
This is because a Muslim woman is not allowed to be married to a kaafir at all, because Islam should prevail and not be prevailed over.
So become Muslim, and present Islam to him (your husband). If he becomes Muslim too, then praise be to Allaah. Maybe this will make him give up alcohol. If he does not become Muslim and the iddah time expires, then leave him, especially as his circumstances are not acceptable and and his alcoholism is causing so much trouble for you and your children, and living with him is unbearable. Do not feel any regrets about him; maybe Allaah will compensate you with someone better than him. If he cannot take care of himself, then he only has himself to blame, because he is bringing it upon himself. If you leave him, this make him feel bad and thus bring him back to his senses; maybe then he will become Muslim and get his wife and children back.
The second problem is very serious and dangerous, because it has to do with an Islamically unacceptable relationship between a man and a non-mahram woman who talk together intimately about all kinds of topics, without any controls or restraints. This improper conduct has led to other kinds of improper conduct, namely love and attachment and fear of separation from that person, as well as the weighing up of this relationship against the religion of Islam even though continuing this intimate relationship is so harmful and entering Islam is obligatory and will bring happiness in this world and salvation from the Fire in the Hereafter. Moreover, we are astonished and wonder how he can be so committed to his religion as you say yet he has formed such a relationship with you that you describe him as your friend. You must hasten to enter Islam and advise this person in an appropriate fashion (such as sending him an e-mail, for example). Rest assured that when you become Muslim, Allaah will make for you a way to get out (from difficulty), and He will provide for you from (sources) you could never imagine. So strive to please your Lord, and He will be pleased with you and will make other people pleased with you. May Allaah help us all to do that which He loves and which pleases Him.