Dear T.:
Children take things that their parents say much more to heart than we realize. Also, your husband is likely caught up in the conflict of the moment and probably does not realize that what he is saying is very hurtful to your child. Children need to know that their home is a safe haven and, when a parent threatens to leave, this disrupts the child's sense of well-being. My advice regarding the fighting is to refuse to fight with him until you are alone, if you have to fight. Don't buy into the verbal conflict. Say "This is not an appropriate time for us to discuss this matter. We'll talk about it later."
You seem like a very busy person. Does your husband also work? If so, part of your problem may be that he feels ignored or that you don't have time for him. From experience, I know that this is an issue for some men. Also, the fact that you work three jobs may also have some bearing on his feelings about being able to provide for his family and thus reflect on his self-image. I am not laying blame on you for the conflicts -- just trying to let you see the situation possibly from his point of view.
As hard as it is to do, you are probably going to have to rethink your scheduling to include some "together" time for the two of you. Maybe you could trade some kid-sitting time with a friend, or arrange for your kids to sleep over at friends' houses so the two of you could have a date. Men apparently need to know they're appreciated. Mine had this really big thing a few years ago that "he wasn't appreciated and nobody thanked him for the things he did and everyone just took him for granted." I had no idea that this was how he felt, and so made an effort to thank him, compliment him and make some time for us to be together by ourselves. It made a big difference in our relationship, and thus for the whole family. I know it is hard to do, but someone has to be the adult and, if it makes a difference, it's worth it.
Regarding your kids, they need to be reassured that you both love them and that, no matter what happens, you will both always be there for them. You could tell them that sometimes adults argue and say things they don't mean. I believe it's also important to let them know that they did not cause the conflict and that it is a thing that you and their dad have to work out together.
Hope this helps some. Please accept my deepest sympathy on the loss of your Mom.
E.