I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear that this is happening.
A 30 day notice delivered on August 3 is not effective for September 1, because that's not 30 days. New York is not one of the states that requires 60 days notice for a rent increase of more than 10%, but she still has to give you 30 days notice. A rent increase can never be retroactive, and the landlord did not give sufficient notice to increase the rent as of September 1. You'd only have to pay $650 if you stayed past October 1, and she's already requested that you be out before then. She cannot insist upon the additional rent for September or October, and she cannot evict you for refusing to pay an illegal rent increase.
If you're in New York City, you may want to call the city at 311 to see if there's anything they can do to help. The Marshal will not come to the apartment to remove you without a court order, and she cannot get a court order before September 1. (She shouldn't be able to get one before October 1, either.)
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