What can we do if our landlord claims that we didn't provide sufficient notice to vacate when we did and now they're charging rent fees?

Question Details: We asked our apartment office what the NTV process is when we were waiting to close on a house. She said 60 days notice, and 85 relet fee. So later, when we closed on our house, we asked via email how we can issue our 60 day NTV. The assistant manager replied our move out is 7/14 and asked us to come in to fill out the 60 day NTV but we were out of town due to death of a family member and asked if the form could be sent via email. We didn't receive a response so when we came back into town we went into the office to confirm the we would be charged $85 fees and our move out date of 07/14, and we signed the form. They confirmed with us verbally our notice date from the email would be honored. I received email confirmation from the assistant manager that said 07/14 was entered into their system. We paid our fees and prorated rent through their automated system, which automatically calculated through 07/14. They no-showed our move-out walk-through. We called for 2 months for our final statement and deposit refund. They now say we owe concessions accelerated rent in addition because "we failed to give notice and our move out was set for 07/31". Can they do this?

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