YOU DO have the right to be happy, of course you do! We all do!
But you have to do it right, you have to be brave and you have to face what needs to be faced.
Yes it'll be hard before it gets better, but at least you have hope that it will get better. You stay in this sad relationship, where is the happy ending? If it's not there, it's not going to be there until someone does something about it.
You don't want this relationship, GET OUT. It really IS that simple, you see what you need to do, you communicate it, and you action it.
Not saying it won't hurt, isn't sad, it will and it is, but an unhappy mother is not as great a mother as a happy one is.
You get yourself and your DC to a happy home, perhaps find a new partner that IS great in time, and it will work out.
You can't suffer because of the thoughts of what a child might go on to think.
For all you know, they might be happier for you when you are out and glowing.
I don't believe that someone like you would have brought her children up to think that of their mother.
You need help communicating stuff about all this to them? just ask. many women here can help you explain it to them so they will understand.