Who is responsible for taking care of my blind and sick cousin?

It is not mandatory that the adult children take care of him. You have to petition for guardianship otherwise he will become a ward of the state. Use of a lawyer is recommended.

Yes, his kids should be taking care of him. But they cannot be forced to do that. See about the benefits available to him through the government, like medicare or medicaid, whichever it is that he can get, and local government assistance. Perhaps a local social worker can help the most.

It's somewhat unclear whether you are asking about not having to take care of your cousin any longer, or how to make more legal your care of your cousin. Generally, in FL there is no legal obligation that adult children take care of their elderly parents. Accordingly, there is also no legal requirement that you, as a cousin, do so either. Generally, the Dept. of Children and Families or the Dept. of Elder Affairs would have jurisdiction to place your cousin into care if you no longer want to provide it. If you want to make your status as a care provider more of a legal role as well, you can file pleadings with the court to be appointed as the legal guardian of your cousin. You may also want to consider having your cousin meet with an attorney to create or modify their own estate plan, particularly given the condition of their health you described. Consult with an attorney and discuss the specifics of your situation and your options.