Depends on whether you have accumulated the required amount of work credits. If you did, then the sponsorship requirement is over. Also, once you become US citizen, the requirement will also terminate. If you are eligible for US citizenship, then I highly recommend that you apply. Consult experienced immigration lawyer to assist you.
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Usually it is the divorce court which would enforce any affidavit of support. If not, then perhaps you could try a contractual lawsuit. Note that under a contract suit, you have a "duty to mitigate" which means that if you can work and support yourself, you can't make him support you.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Your question is an immigration/family law crossover. Your facts indicate that you were married for at least ten years and this would mean it was a marriage of long duration and you waiver spousal support in the judgment(divorce papers).
As to the immigration question: your husband signed an I-864 Affidavit of Support as the Petitioner and he is financially liable until you become a citizen or earn 40 qualifying quarters of work in the United States which is generally 10 years. Divorce does not end the sponsorship obligation if this has not been met.
YES. Your husband has an obligation to support you based on basic state family laws that allow a spouse to recover spousal support from her former spouse through a typical dissolution of marriage (or even after the marriage has been legally dissolved, if the court overseeing the divorce retained jurisdiction to determine spousal support). Additionally if you became a lawful permanent resident through your US citizen spouse no doubt he signed an affidavit of support promising to support you so that you would not become a burden to the US or the any of the US states. That is a binding contract which obligates him to support you.