Yes, that wold be true --- but only only if we were talking about Actual Sin, the sin one commits.
It is not true of Original Sin, the sin one inherits.
At the beginning of their lives, both Adam and Eve, in Eden, enjoyed Original Innocence, Original Justice, Original Blessing, Original Grace. Despite all this, they sinned. They still had free will. So would Mary have had free will. Otherwise we are all nothing but ventriloquists' dummies, sock-puppets, androids, and this world is the Matrix and whole shebang is a farce.
Scripture tells us that we are slaves to sin.
We don't have free will.