The first amendment is pretty specific: "Congress shall make NO LAW abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble." By congress they mean the government. Any government, state or federal.
No law means no law. It's similar to wording as the second amendment, which says "the right of the people...shall not be infringed."
So they want full freedom with respect to guns, but not to speech. Anything that stands in the way of the press doing its job is against the 1st amendment. Making it easier to sue places the burden on the press to defend and protect itself.
There's another Florida bill requiring bloggers who talk about the government to register with the state. Registering with the state is abridging the freedom of speech and the press.
Is it the end of political talk shows? It might if one gets sued. That creates a chilling effect, and could lead to radio stations creating rules about what their employees can say on the radio. Employers can limit free speech. The constitution only says that the government cannot. In this case, the state government is imposing rules that would be viewed as unconstitutional. It's not the first time. A lot of these state laws, dealing with morality, diversity, or education, are unconstitutional.