To provide some factual information, it is important to understand that a person who provides false information to a lawful enforcement officer has the potential to carry substantial jail time and fines. It is understandable that you wish to remain in the United States but by providing false infomation, you may be turning a minor ticket stop to a more substantial charge.
In essence, the risk of leaving the United States may not be your only concern so it may be a good idea to advise your attorney as to your situation to determine if you can disclose the error prior to law enforcement matching you with the federal database and prior deportation order. If the authorities unravel the truth, you would probably have a much more difficult road ahead of you.
Unfortunately, there are not really any good options for you in this situation but continuing to misrepresent the truth after your initial fear has subsided will probably not create a better situation for you.
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