Congratulations on such a strong match. You and this woman are close family.
She is not your grandmother. The cM total is out of range, the age difference is unlikely, and she would know if she had a child that was raised by someone other than her.
She could easily be the full sibling of one of your father's parents. That's your great aunt. While many people who were given up for adoption turn out to be the half siblings of other children either of their birth parents had, it is not uncommon for a full sibling to be given away for adoption (or taken away from the family, or lost, or even switched at birth, though that one is fairly rare). Sometimes this happens under severe financial distress.
She can also be your father's half sister, your half aunt. With a 12 year difference between them, this is just as possible as being your great aunt. If this is the case, then your father and her would share a mother (not a father). Why? Because your father did not get any part of the X chromosome from his father and therefore could not have passed it down to you, his daughter. Your match would have gotten X from both of her parents.
If she is your half aunt, you can imagine that your grandmother had a child young, probably out of wedlock, gave her up for adoption, then went on to marry and have other children, including your father. This is a possible scenario and far from the only one.
The only other possibility given the numbers is that this match is your first cousin. This one is a lot less likely because there is a 30-40 year age difference between you. When you say "she was born 12 years before my father if that matters," the answer is that it does matter and I'm really glad you gave that bit of information. It doesn't eliminate the possibility of first cousin, but it makes it almost impossible.
In this case, your match would be the daughter of one of your father's full siblings (could be either a sister or a brother). This could only happen if your grandmother had one child very young and then had your father when she was much older, with the same man. This happens. Imagine your father's much older sister having a baby as a teenager and giving her up for adoption (or his much older brother impregnating a girl as a teen and either giving the baby up or not even knowing there was a baby).
Your next step is to find other cousins to test and see if they match her, and how strongly.
At the same time, take the step of working the paper genealogy. While she doesn't have a birth certificate stating her biological parents (though these are not impossible to find), she will know where she was born and adopted from, which religious organization (if any) was involved, and a few other facts that can help you narrow things down. I recommend as well that she get a search angel, which is a volunteer who helps adoptees find their birth families. Depending on which state she was born in and adopted from, it can be possible to find documents.
