This sounds like a more complicated situation than the question lets on. You would be well served by speaking with an immigration attorney and an attorney who can handle international child custody issues.
Your situation is more complicated than this. There are serious immigration and criminal penalties for fraud. You need to speak with an attorney.
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If the attorney who has been helping you and your husband does not know the answer to this, you need a new attorney. There are several different options available to you. However, you have to act quickly before you could potentially triggers grounds of inadmissibility that could bar you from the U.S. A specific answer cannot be given until an attorney has a chance to review all of the circumstances in you and your husband's situation.
Not all states recognize marriages between cousins, so you will need to settle in a state that permits that type of marriage. Also, you will have to prove to immigration that your marriage is bona fide so you should not marry unless intend to establish a life together with your future spouse. Unless your future spouse obtains U.S. citizenship, you will need to maintain status in the U.S. to be able to apply for a green card. If you do not want to leave the U.S., you will need to find an...
I agree, you need to speak with an attorney. You need to navigate a strategy that takes into account your immigration goals along with preserving your and your wife's current status and that also takes into account the needs of your current and prospective employers.
You have a good problem to have - you are potentially eligible for two different nonimmigrant statuses. Navigating the process so that you maintain status without inadvertently jeopardizing H-1B application requires careful legal work by an experienced attorney. You and your employer should contact an attorney as soon as possible.