Apartments Approved My Unit To Unit Transfer, But Now Want Me To Pay Rent On Both Apartments

I moved out of one unit to another back in September. I signed a new lease, a transfer agreement, a notice to vacate, along with other documents before getting the new keys. They even transferred my old apartment deposit from one unit to another.

I payed rent for September for my new unit ONLY.
and only accidentally payed rent for both units in October.

My apartments claim i never surrendered the old apartment because

1. I never returned the keys (i wasn’t told i had to)
2. I payed rent in october ( which i already explained to them that was not knowingly. And i also questioned that if they thought i remained in the old unit in September and October why didn’t i have to pay rent in September? wouldn’t i had been given an eviction notice??)

I pulled up my lease and showed them that a notice to vacate was proper surrendering of the apartment and turning in the keys was not a requirement.

But now they claim i never signed a notice to vacate BUT I DID! (i do not have a copy because it was in person and on a pre-printed form that they created and I just signed. I wish now i would’ve taken a picture of it ????‍♀️)

Furthermore signing a notice to vacate is literally a transfer requirement according to their terms and i literally have an email of them telling me “Hey when u get a chance come down to the office to sign a notice to vacate because that is required to proceed the transfer process”

But since they claim i never signed one they’re refusing to refund me for October and continue to charge me for November

The whole transfer process took 5 months in total they had so many times to tell me if i was missing any documents. I even have an email of me asking if there was anything else i needed to do and they failed to notify me of any missing documents. On move out day i was there in person and they even had me sign even more paperwork ????‍♀️

They acknowledge that yes the transfer was approved and also the transfer of my deposit from my old unit to the new one, but “that doesn’t mean that my old lease was terminated”

What should i do? Is this worth taking to court? Or are my chances of winning too small?

Location: Texas