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You need to get the J-1 waiver, then you can apply for the green card. You can not get the EAD unless you get the J-1 waive to file I-485. marrying a US citizen is irrelevant to EAD application and is not helpful to J-1 waiver either. AS long as you can get the J-1 waive, you can apply for green card in US even if you have been unlawfully present for a long time.

But if you leave US, then you will be subjected to 3- or 10-year bar if your overstay is > 6 months or > 1 year.