I too, am an alcoholic. I have been in recovery for about 4 1/2 years. This is NOT a man you should legally tie yourself to. I promise you will be miserable if you do. That said, all we can do is urge you to do the right thing for yourself and your children. Living with an active alcoholic is torture and your children know more than you think. They may not know exactly what they know, but they know something is definitely wrong.
You said that if you gave him an ultimatum, he would show you the door. That should tell you something. The fact that you are afraid to make him choose is what he is betting on. Is being married so important that you would risk the livelihood of yourself and your children? Marriage can be difficult under the best of circumstances, but when one partner is an active alcoholic, everyone is miserable.