Can the Management of Apartment Pay the Moving Charges and No Lease Termination Fee as They Are Not Able to Fix the Problem?

You need to document the problem first by sending the landlord a certified letter indicating your concerns. This will give them the opportunity to mitigate/fix the problem. At the same time, you should call the building inspectors office of the city in which you live and ask for an inspection. If the landlord is in violation of the minimum codes they will cite them with a fine. This will further help your case. They may also require them to immediately fix the problem.