The distinction that I think you are making is the distinction between "procedural due process" and "substantive due process." For a period of more than 40 years following 1890, the Supreme Court declared unconstitutional many state statutes which were designed to regulate businesses or to establish minimum standards of health and safety in the workplace. Lochner v. New York (1905) (in which the Court held that the attempt by the state to establish a limit to the number of hours an employee could work as a baker was an unconstituional deprivation of "liberty" in violation of the Fourteenth Amendment) is one of the better known examples of that particular series of "substantive due process" decisions. At or about the time that FDR began talking about the need for more than just nine justices on the Supreme Court, the Supreme Court pulled back and then began finding ways to approve of regulations on business.
More recently, "substantive due process" has returned and has played an important role in connection with issues relating to what are often termed "personal interests." Roe v. Wade (1973) is probably the best known example of this excursion into the realm of "substantive due process."
Do you believe that the state of Texas could, consistent with the Fourteenth Amenmdent, make it a crime to "make any false or critical statement concerning the governor or any state legislator"?
The only place I could see them running into trouble on that is if there wasn't a clear understanding of what constitutes "critical". Then, we'd have the same situation I talked about before - "void for vagueness". In order to have due process of law, you need a law. In order for it to count as a law, it has to have a discernible meaning. Otherwise, it would be a little more than a license for a public officer to rule by decree, which defeats the whole purpose. Other than that, I don't see any due-process violation in such a law.
Just out of curiosity, did Texas actually try to pass what you just described?