Refiling is usually quicker. Make sure you can cure whatever was the basis for denial. I can't tell by your posting if he is an immigrant or born in the U.S. If the only reason for denial was that you didn't provide evidence of his USC, and you now have his U.S. birth certificate, then refile.
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One always needs to exhaust the appeals. Filing a new VAWA self-petition will not resolve the underlying issue, whether it is an issue with good faith marriage or the presence of abuse. You must retain a competent immigration attorney who can sift through and identify the pertinent issues in your case and advise you regarding the appealable errors of facts or law made by USCIS.
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Not sure what went wrong and whether it was USCIS's error or yours; birth certificates do not say people's immigrant status so I don't understand the relevance about what his birth certificate doesn't state. Generally, if USCIS was correct in its determination and you can re-file to correct that error then re-file. If USCIS was wrong either with facts or law then an attorney should be able to discuss with you the pros/cons of re-filing or appealing.
Hi hope that you are well. You should go see an attorney and get a consult. Make sure that you take the denial decision to her so that she could review it. Based just off what you are saying I would lean towards appealing but please remember that there is a deadline for that so please don't delay. Good luck!
Talking to an immigration attorney would be best.
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