I think you need to have a sit down with your Dom and tell him exactly how you are feeling. He needs to negotiate the time spent with you and the new sub.
If you are reacting so strongly against this, maybe you aren't best suited to a relationship in which you need to share.
I've very lightly dabbled in adding another to my relationship and hated it. I want all of my Doms time and attention and hate seeing him with anyone else. There's nothing wrong with this.
Remember that even though you are a sub you should be getting something positive out of your relationship.