I got accommodation offer in November and now I am worried

Hello and good day.
I am an international student at the University of Essex. I had gotten and accepted accommodation back in November 2024 for a studio, which I forgot about. Come this year, I got a good apartment off campus for a good price, and I was forced to move due to a bad experience at the uni accommodation, the price, as well as my sponsor’s request after a few incidents.
I received this apartment a few days ago and the accommodation team even sent in a reference as well, but out of nowhere I was inducted into their studio without paying any deposit or doing the induction manually. I was very confused, and when I called them, they told me that I did the induction on the 18th of September, the same day they sent in the reference to my landlord. It was very confusing, and they told me I had already confirmed a place with them last year and there is no cooling-off period. So I sent them an email explaining the situation and telling them I am willing to comply and do what is needed.
Also, they have raised the prices of the accommodation, making it extremely unreasonable, and I don’t want to be liable or find a way out of the contract. I am worried that they will tell me I still have to pay them in full despite not paying any deposit or anything.