What to do if my Landlord is denying me to switch apartments.

Every apartment is bound to an "implied warranty of habitability". This means that the apartment must be clear of defects severe enough as to make inhabitation of the space dangerous or impossible. A leak in the cabinets could cause mold which is dangerous to your health. The Landlord is responsible for making these repairs in a timely fashion or providing you with a suitable substitute. Since you have already spoken to the Landlord and they have agreed to let you move I would suggest talking to them again and setting up a date for that move. Due to the fact that your Lease is up at the end of the month they may ask you to sign a new lease.  In the event that you stick it out to the end of the lease and then find a new apartment you may want to consider asking for a refund of your last months rent as you were living in an uninhabitable space. I doubt that you will be able to make any headway on this without engaging a lawyer, but it's worth a shot. If you would like to take action on it then you likely will be best served by making a claim in small claims court. There you can represent yourself without a lawyer and will have minimal filing fees. You should list your damages as being the rent for any months that you were affected by the situation, any repair, medical, or other costs related to the damage. If you are being forced out due to the damage, you can also include the cost of searching for a new place to live.