If it were me . . . I would keep the letter I received from the property management stating I could move out without incurring any fees close to my chest, especially since I like surprises, more importantly surprised looks. When a property management company forwards to a collection agency those files they believe have a positive balance, bad things tend to happen. So many bad things that in my firm, I do not take on that type of work.
The laws which govern debt collection frown upon the misrepresentation of the nature or extent of the debt and to the extent that this collection company is attempting to collect rent which the management company forgave in that letter, they are misrepresenting the nature and extent of the debt.
Since I do not know how much they are demanding or your feelings toward satisfying those demand to avoid litigation anything I say beyond this point would be pure speculation.
A proper response would require a thorough investigation into the history and background of this relationship. The information provided above is just that, information, to be used as you see fit.