Is the woman considered legally married or an adulteress after she said to the man who wanted to marry her: "In the name of Allah, I marry myself to you, and Allah is a witness to what I say," given that the marriage has not yet been legally documented?

It appears that the offer and acceptance between you occurred without a guardian (wali) or witnesses, which renders the marriage invalid. You must sever all ties with this man because your relationship is forbidden, and any intimacy between you is considered pure adultery. Repent to Allah. If he also repents, then it is permissible for you to marry, provided there is a guardian, witnesses, and a dowry. It is not a condition for the validity of the marriage that it be conducted through an official authority.