[TX] 60 days written notice

My lease contract with my landlord (part of Texas Appartement Association) is requiring a 60 days notice. But one paragraph says : « your move-out notice must comply with each of the following : (c) 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 send a 30 days notice on may 2nd to leave on June 6th which is the last day of my initial leasing contract. They want me to stay until July 2nd.

My landlord send that written reminder on April 17th which is out of the window to send it (too late)

I want to pay to June 6th and let them (maybe) sued me for the upcoming month (about 2000$) but is it worth it ? I m French and don’t know anything about court but I think I m right following my contract.

Is it expensive? Can I defend myself without an attorney ?

Thanks