I came to the USA on a visa waiver program. I have known this lady for the past 4 years. She is a green card holder. I do not want to be in trouble with the US immigration.
You cannot file for adjustment of status until your priority date will become current (~17 months after filing), and, if you remain in the U.S., you must maintain legal status until the date of filing for adjustment of status. Since you entered on a visa waiver, you cannot extend or change your status. Therefore, you have to go home not later than on the 90th day from your entry. If your fiancee can become a citizen soon, and you wish to become a U.S. permanent resident without returning home, you should discuss your case with an immigration attorney face-to-face. Otherwise, after your marriage, you should go home, and your wife would file an immigrant petition (Form I-130). In about 17 months after filing, if USCIS approves the case, it will be transferred to the National Visa Center of the Department of State. NVC will send your wife some forms to fill and visa invoices to pay. After receiving all they request from your wife, NVC will send the case to the U.S. Embassy in Wellington. The Embassy will invite you to an interview and, if you satisfy their requirements (police clearance, vaccinations, etc.), issue a visa allowing you to enter the U.S. as an immigrant. After the entry, you will have to apply for a 2-year green card conditional on your marriage (if it falls apart, you will lose your green card). 18 months later, you will have to apply for "removal of conditions", i.e. for a permanent green card no longer tied to your married state.
More information is needed in order to analyze your and your fiancee's situation and determine the appropriate steps to take. You are authorized to remain in the U.S. in compliance with the Visa Waiver Program, or, if you are eligible for and timely obtain a change of status, then you will become authorized to remain in the U.S. only in compliance with that new nonimmigrant status. After becoming married to your Permanent Resident fiancee, she would be able to petition for you in the Family-based Second Preference (F2A) visa category. There is a backlog for visas in that category - according to the November, 2015 Visa Bulletin issued by the U.S. Department of State, visa numbers are authorized and available in this category only for people who have a Priority Date (filing date) of May 15, 2014 or earlier. Once a visa becomes available, you would be able to apply for adjustment of status in the U.S. to become a Lawful Permanent Resident (to get a "Green Card") if at that time you are lawfully present (present with a valid visa); alternatively, you would be able to apply for consular processing from overseas. A different analysis applies if your fiancee soon may be eligible to apply to become a naturalized citizen, since (1) there is no backlog for visas in the "Immediate Relative" visa category, and (2) if you entered the U.S. lawfully and with inspection, the failure to maintain status (such as overstaying lawful presence) would not interfere with eligibility for adjustment of status in that category. Especially since you are less than fully familiar with the immigration application process, it would be wise for you and your fiancee to work with an immigration attorney who, after learning all of the relevant information about you and your fiancee, would be able to provide advice about immigration eligibility, options and strategies. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.
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