The answer to your question is provided in post #1.
Is CA 'legally' banning assault weapons by ignoring our 2nd Amendment?
The First Amendment, as written, placed no restrictions on state governments. Although its plain meaning would forbid Congress from passing laws restricting broadcasting or campaign advertising, nothing in the First Amendment would forbid states from imposing even draconion censorship laws if they so choose.
One fundamental difficulty with the notion of "incorporation" as applied to the First Amendment is that while the First Amendment is vast in what it limits (it forbids any sort of law that might abridge the enumerated freedoms) it's narrow in who it restricts (Congress), the protections states must give are necessarily narrower in what they restrict and broader in who. Trying to apply the First Amendment to the states is thus a very poor fit.
Or, to look at things another way, the First Amendment should be applied against the states, but only in a literal fashion: no state should a abridge people's right not to have Congress of the United States pass laws restricting the Freedom of the Press, Freedom of Religion, etc. Pre-17th-Amendment, such a rule might actually have had some meaning, as it would forbid states from appointing Senators who would seek to pass such laws. And I suppose even today it could be applicable against Electors, though--sad to say--such an interpretation would forbid states from appointing electors who would vote for Bush.