I am pretty active on this site today--haven't been able to get time like this for a while. Here is my question: 

How do you handle a conflict when you have young children who are around, or even in the room? 

Sounds almost ridiculous, so to be more specific, how do I handle it when my husband does something that I find unacceptable or unhealthy or unkind, in front of our children? 

Example: 

I get up first every, single, day. After years, I have given up on the idea that I will be able to sleep in or not be woken on the weekends by my children who need me. We have no family or help nearby, it is just me. I work really hard as a teacher. I am so tired that I don't even feel human most of the time. My husband sleeps in a different room (long story). This week, to discipline one of my 8-year-old children, who hit his brother really hard, I gave him a consequence of no video games, for a week. Let it be known that I am almost 100% in charge of our children at this point, long story. My Dh is sick and on disability right now.

Well, we were on day 6 of the consequence this Saturday morning, and son has been doing very well, and I am so, so, so tired, that when he tried to wake me at 7, I told him that he could play a (safe) video game while I slept for one more hour. Well, I overslept, and around 9, my DH got up, atypically, and I heard him questioning my child, "so, your mom went back on her word, huh? She told you a week, and then changed her mind, and now it is only day 6, and she is letting you play? Again?" (Evidently I have done this before. I know it is not great, but every once in a while I will modify an consequence if the point has been made or improvement is shown and my motivation this morning is that I just wanted to sleep for another hour or two).

I heard, went into the room and just said to DH, "Please stop." I was just so mad. I had tears in my eyes. His choice of words, his painting something negative about me to our child, his public discussion in front of our children, in criticizing me when I do almost all the work while he sleeps. I got sucked in, and started arguing with him when he replied, "I didn't mean anything by it, you are too sensitive. Don't even talk to me for the rest of the day." Sorry, but with his choice of words, I didn't believe him, and I don't think he believed it, either. And it is his tactic to cast me as the bad guy who is too sensitive and wants to argue all the time. 

We are not in a great place now, I am thinking of separating, but I want to protect my children and I don't want them to be damaged or to complain about their father to them. I prefer that all conflict and serious discussions between spouses be handled in private but my DH thinks nothing of doing it in front of them, even when I ask him to stop. What do I do?