I think honesty is best. I was raised by a single mom and she always let us know she was dating. When we got older (mid teens) we knew more about her sex life, but we had an idea about her being "romantic" before then. I remember I was 9 when I found out my mother was pregnant with me when she married my father, and I was upset -- but I think the fact that it had been treated like a deep dark secret played into that.
I don't know, maybe this is why I have always chosen to be childfree, but I would not let children "run" my household with their hysterics. Personally, I think children today are way too spoiled and treated like such fragile beings. Yes, get them counseling if needed, and be compassionate, but I think it's still important that they face reality and not live in illusions.
If I were you, I would sit down with all of them, and your wife, everyone together and explain that this is "grown-up stuff" that they won't completely understand, and they don't have to like it, but it is your personal life and one important thing they need to know is that your having a gf is about having more love in your life, not taking it away from them or your wife. I would give them space to voice their fears and concerns and address the ideas they have about what a family is, but really, they have to stop throwing tantrums and feeling sorry for themselves. There are kids out there who have much rougher stuff to deal with - maybe have a family "count our blessings" session every night before bed, and everyone talks about things they're grateful for. It couldn't hurt.