A couple points:

Did you document the spots on the carpet when y ou moved in? If not, that's a tough break - then you can have them inspect the carpet on the way out as well.

Not sure what your lease says, but usually you need to give a 4 week notice. They can not keep charging you rent - unless you're breaking the lease early.

Better to work out an arrangement with your landlord (perhaps pay for part of the carpet cleaning), than to have issues with leaving and additional potential credit issues.