Adopted Mother

Hello Everybody! I am so happy that I stumbled upon this community. I have recently discovered that I have a couple of Luxembourg Ancestors! I have long desired to become a dual-citizen and never in my wildest dreams had I thought that something like this would arise. Upon learning about it I began to look into the possibility and read hardcore. But I haven't been able to find anything regarding adoption, maybe I haven't looked hard enough. Very recently my mother was able to find out who her birth mother was, which is my link to the Luxembourg ancestors, because of a law that was passed in Washington she could receive a birth certificate that wasn't certified and was only a copy, it said, "This is an original record from a sealed file. This is not the legal birth certificate currently on file for this person." (This is the original birth certificate and I might be able to get it certified, but read on). Now my huge dilemma arises, I would like to claim ancestral citizenship and my mother would like to also if there is a way. The issue comes into play with the fact that her current name is not on the birth certificate, only her mother's full name and my mother's last name. My mother was brought into the home of her adoptive parents 4 days after birth and it was made official 5 months after birth. Her certified birth certificate has the names of her adoptive parents on it. I seem to be stuck on what to do. I can gather all the other pieces of info no problem, but I am waiting until I can figure this dilemma out. If anybody has some insight, please help Thank you so much! 

-Christer