60 Day Notice

TX - I am planning to move out of my apartment at the end of my lease term (early June). I wrongfully assumed that a 30 day notice was required which I attempted to give today. It turns out my lease states a 60 day notice is required. I know that if this was the end of the story, I would be on the hook for the rent.
Further reading the lease, however, there is a clause which states: "If we require you to give us more than 30 days’ written notice to move out before the end of the Lease term, we will give you 1 written reminder not less than 5 days nor more than 90 days before your deadline for giving us your written move-out notice. If we fail to give a reminder notice, 30 days’ written notice to move out is required."
I never received any notification in the mail nor by email stating that my 60 day notice deadline was approaching. I called the leasing office and they are arguing that the lease renewal email contained a reminder about the 60 day notice (it did not) and that they taped a paper notice to the door of the apartment (which I did not receive).
What is the best steps moving forward?
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