As long as you entered the US lawfully and DO get married to a US citizen, he will be able to petition for you and you will be able to adjust status to permanent residence even if this is done after your I-94 stay expires. However, if you do not get married and overstay, then you will be facing potential bars to reentry to the US once you do leave the US.
Your question requires an attorney consultation. It is not a simple question that can be answered on this type of forum. There are many factors that would need to be considered and evaluated. I strongly suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case and advise you of your options.
If your fiance is a U.S. citizen and, after marriage, petitions the government for your green card, overstaying your admission will not disqualify you from adjustment of status in the U.S. If he is not a U.S. citizen (a lawful permanent resident, or an alien living in the U.S. of a non-immigrant visa, or a person without a legal immigrant status, you should get a consultation with an immigration attorney to have a detailed discussion of the options that might be available to you.
You should both consult with an immigration attorney to fully review your options prior to your overstaying your I-94. But typically, yes, he will be able to petition for you if you do overstay.
You can apply for a green card as a spouse of a US citizen if you lawfully entered the US. You must not be in status in order to apply. I recommend that you consult with an immigration attorney if you do not know how to go about it.
You can overstay your 6 months and marry down the road. And then apply for your green card. There is no penalty for the overstay. And the entire process if done correctly takes only about 3 months from start to finish.
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