Can an employer sponsor me for work authorization or adjustment of status?

If you are out of status you cannot adjust in the U.S. Therefore likely the petition through an employer will not result in residence. An exception is if you were the beneficiary of a petition prior to April 20, 2001.

Asking about employer sponsorship is premature, since the first issue to be addressed is the consequence of your years of unlawful presence in the U.S. When a person is in the U.S. without valid immigration status, in order to apply to adjust status (to apply to become a Lawful Permanent Resident or get a "Green Card"), or even to apply to change status (to apply to get a different non-immigrant status), that person must go outside of the country to "consular process." Under current immigration laws, someone who has been unlawfully present in the U.S. for more than a year, however, will be subject to a very harsh 10-year bar to re-entering the U.S. In relatively rare circumstances, one may apply for a waiver of the 10-year bar. To succeed, generally one must establish exceptional hardship to a U.S. citizen spouse or parent.