Carefully review your lease agreement to see if it addresses such situations. If you don;'t have a lease or if the lease you do have is silent on this matter, then you are not responsible for the increase in premiums.
Under an Oral Lease, you are still responsible for the damages that you caused. The landlord's hurdle is going to be proving that the property was damaged when you moved out. Because of your conversations at the end of your lease term, this is going to be a classic "He said, she said" type of case.
First and foremost, you need to review your lease for any language that addresses this situation. If the lease is silent, then the landlord does have the right to make reasonable repairs and bill you for the same.
If you disagree with the "reasonableness" of these repairs, you can withhold payment. However, if you withhold payment, you are opening yourself up to a potential lawsuit which could result in an eviction.
I suggest you contact a local Landlord Tenant Lawyer to review this...