Where is the constitution does it say the state can't do that?
The 9th makes it clear that not all rights need to be explicitly in the text.
As for the content: the law says what it says. Where in the constitution of the United States of America and its amendments does it say that a state government must be truthful about public health statements?
The law says what it says. A behavior might be rotten, evil, illegal under other documents, and an all-round bad idea and still not a violation of, in particular, the constitution of the United States of America and its amendments