Is a Shariah-compliant divorce necessary after an American court divorce, based on the husband's committing adultery with the wife's mother?

Your husband's adultery with your mother is one of the gravest sins, as it is incest. It was narrated from Ibn Abbas, may Allah be pleased with them both, that the Messenger of Allah, peace and blessings be upon him, said: "Whoever commits intercourse with a forbidden relative, kill him." Both of them must repent sincerely. Scholars have differed regarding whether the wife becomes forbidden to her husband if he commits adultery with her mother. The more cautious approach is for the spouses to separate. If your husband does not divorce you, you are still under his marital authority, and the divorce issued by the court is not sufficient. If you follow the opinion that states she becomes forbidden, or if you dislike staying with him, you can refer the matter to Islamic centers for them to separate you.