First, I'd recommend not getting anymore dogs (or other animals) until you move out on your own away from your mom! She obviously thinks dogs are just objects that you can throw away when you're tired of them. They are living things and do not deserve to be treated like that.

I'd try talking to her and convince her not to get anymore dogs herself. Write a list of all the dogs she's thrown away and ask her if she even still keeps in contact with ANY of the new owners. If she doesn't, she has no way of knowing if the dogs are okay or not.