I am a mainstream, normative Orthodox rabbi, and I say to you, with sensitivity and with honesty — because your question reflects that you want to hear the truth and not puffery that merely will make you feel good about yourself for a few minutes — that mainstream, normative Orthodox Judaism does not recognize the child of a non-Jewish birth-mother as a Jew unless that child converts to Judaism in accordance with the standards of Orthodox Jewish law. Thus, your status as a Jew is not related to whether your husband is Jewish, how you rear your children, whom they marry, how you feel or identify, or anything other than your birth mother. If she was not Jewish, then you are not. If you are not, that means — because you are the female —that your children are not, and their children will not be. And someday, long after we all are gone, some descendant will learn, “out of left field,” that (s)he will have to go through the process of an Orthodox conversion in order to marry the Jewish person whom (s)he loves and who loves her or him, because no one previously in the family tree underwent that conversion process.
You can change that “Ghost of Chanukah Future” and can assure your lineage its Judaic authenticity for all time to come. To do so, you would have to undergo an Orthodox conversion.
I have been involved rabbinically in situations like yours, and someone with your situation presently is in our shul’s conversion program. For an overview of our shul’s and my approach, please see: http://www.rabbidov.com/conversion/conversion.htm
Basically, you would need to undergo an Orthodox conversion (“Gerut,” in Sephardic modern Israeli pronunciation; “Geirus,” in Ashkenazic / Yiddish pronunciation). The Rabbinical Council of America (RCA) has coordinated its Orthodox conversion program in America with standards accepted by the Israeli Chief Rabbinate. http://www.rabbis.org/conversion.cfm That helps conversion candidates know that the Orthodox conversion to Judaism they are pursuing now will be accepted by mainstream normative Orthodox institutions in Israel now and in the future. http://www.judaismconversion.org/ Thus, you may wish to contact RCA’s Beth Din of America (BDA) at: geirus@bethdin.org
If you ask any other Orthodox rabbinical resource besides RCA and its BDA to conduct you through a process that would lead to an Orthodox conversion, ask the rabbi to sign the following statement: “I personally assure you and vouch that the process of conversion that our Beth Din offers will result, upon your successful completion of the program, in an Orthodox conversion that will be accepted by mainstream normative Orthodox institutions in Israel so that your future generations will be accepted by mainstream normative Orthodox institutions in Israel.” If the rabbi will not make you that assurance, you may draw the proper inference.
It is heart-breaking to meet people who have gone through a Reform conversion, then a Conservative conversion, only to learn what neither of their previous converting agencies ever disclosed to them that their descendants will not be accepted as Jews in Israel and that, as Orthodox Jews continue their demographic emergence among American Jews over the next half century, will not be accepted as Jews by huge swaths of the Jewish population here either. As such two-time converts approach me for their “third conversion,” the question they ask me, with tears in their eyes, is: “Why didn’t the previous rabbis disclose to me the problems I would encounter in gaining recognition as a legitimate convert to Judaism, here and in Israel?”
I cannot answer for them, but I hope this answer from an Orthodox rabbi has helped and will guide you on your journey to attain authenticity. I wish for you the fulfillment of your dreams and aspirations.