We don't fight in front of our kids. We do argue and bicker in front of our kids, and I think that's healthy. They can see that we do disagree, and that we do work it out. But if it starts to get truly emotional or heated, we try to back off until later. It did happen once, when we were both yelling and then I saw my son looking on, and so I ended it by telling my husband over and over again (no matter what he said), "I'm not going to talk to you about this in front of the kids. I'm not going to talk to you about this in front of the kids." It made him angry, I'll admit, and maybe it even made our fight worse later on, but I think it was better than us yelling back and forth while our son looked on.
My husband and I have a policy that we will not go to bed while we are fighting. We have to work it out before bed. This is HARD. It's really really hard! There are nights when I do NOT want to have the conversation again before going to sleep, and don't misunderstand - things are not all better each time. But we will not go to sleep while actively fighting or actively angry (no silent-treatments, etc.)
One of the ways I cool down is by knowing that my husband has heard me. Even if he doesn't agree, if he can parrot back my argument, telling me what I'm trying to say, then I feel I've been heard. This is a very useful technique that we actually learned from Dr. Phil (back when he was useful, before he became the talk-show host he is now). Talk about this technique BEFORE you are fighting. Then, when you're cooling down, say something like, "I don't feel like you're hearing me." Then make your case in an organized way. He should say, "I understand what you're saying. You want [fill in the blank.]" Then you can say, "No, that's not it, I must not be expressing myself well. I mean ~" or "Yes, but it's more than that, it's also ~," etc. He may not agree. He may even say, "I understand, but I really don't agree because ~." But I find that, even if we still don't agree at the end, knowing that he has HEARD WHAT I AM TRYING TO SAY makes a HUGE difference to how I feel about a fight. I wish you luck.