Florida imposes a duty upon landlords to rent property that is habitable. This includes being pest free. Generally, it is reasonable for a landlord to be given a chance to cure slight problems (1-2 roaches, etc.) but this seems like an infestation. It already sounds like they are in breach of their statutorily imposed duties. They can arguably claim that moving you to another apartment is their cure.
An alternative to getting out of the lease is to place the landlord on notice that you will withhold rent to cure the problem yourself. If you choose that route you need to do this in a very specific manner or you may be served with an eviction notice. Chances are, if you serve them with notice you can present them with an alternative to simply release you from the lease and they will let you go.
I recommend you obtain a landlord / tenant attorney to take you through this process. You can contact me at firstname.lastname@example.org or call me at (386) 873-8422 for a free consultation.
This sounds fairly awful, so whatever you do I hope you find a resolution quickly.
DISCLAIMER This answer is not legal advice nor does it create an attorney-client relationship between any user/reader and The Law Offices of Jimmy Allen Davis, P.L.. We encourage and welcome you to contact us about your legal problems and visit our website at www.lawofficesofjimmyallendavis.com or email me at email@example.com.