Welcome to Just Answer. I am here to help you resolve your tax and finance concerns. Please feel free to ask anytime you need extra help.
I would claim her as my dependent and take the adoption credit. She qualifies in that she has been in your care and support for at least 6 months this year.You have a defensible position if it is ever questioned.
One thing people often misunderstand is that a person's Social Security number stays with them for life. The name on the card may change but the number stays. There are a few instances where this is different but they are rather rare.