Even blatant censorship like the Hayes Code or the Comics Code was never enforced by the government and therefore never in conflict with the 5th amendment. It was a voluntary "certification" manged by the industry itself, which just meant movies/comics not adhering to the code would not get a mainstream audience. So the code was implemented from the writing stage.
The US government hasn't been able to resist censorship entirely. Comedians have been arrested for "obscenity". The FCC will happily go after certain violations in TV and radio. The US government has also censored news broadcasts and journalists.
Bush in particular was very aggressive in censoring the news coverage of his war. Most notably, the flag-draped coffins of dead American soldiers were banned from TV news. During the Regan administration the Justice Department also briefly banned the Canadian film "If You Love This Planet" for being "foreign political propaganda".
There is a federal law on the books against obscenity, but it has never been used to arrest a comedian. Comedians like Lenny Bruce, and Musicians like Jim Morrison have run into trouble with city governments. Bruce was ironically arrested in both San Francisco and New York. Morrison was more expectedly arrested in New Haven.
the Justice Department also briefly banned the Canadian film "If You Love This Planet" for being "foreign political propaganda"
The film was never banned, classifying it as foreign political propaganda meant that before it was shown the audience had to be informed: "This material is prepared, edited, issued or circulated by the National Film Board of Canada, which is registered with the Department of Justice, Washington, D.C., under the Foreign Agents Registration Act."