Was there a "fake news problem" before the Internet era? Were people concerned about tabloids spreading false information to gain subscription numbers?

Absolutely. The book Indelible Ink goes into great depth about what was considered news and what is considered libel. Many newspapers and pamphlets were created by people with a strong political agenda. Many of them reported news that they either knew was false or was based on word of mouth in order to get subscribers. In an era where news was regularly reported weeks or months after it supposedly occurred, there was no real way to fact check anything. Much like today, fake news was written for people who already agreed with the authors and was just a way to confirm biases already created. Libel is very hard to prove because it is hard to prove that something didn't happen or that something is false. The government, especially under British rule, attempted to shut down printers and writers who wrote anti-government propaganda, whether the news was real or not.

A great example of this was Andrew Jackson's marriage to Rachael Jackson. Rachael had previously been married. Newspapers that were pro John Quincy Adams constantly made up stories relating to Andrew Jackson stealing another man's wife, running off with a loose woman, etc. even though there was no real evidence that Rachael hadn't left her old husband prior to marrying/living with Jackson. It may be true that Rachael wasn't technically divorced prior to her relationship with Jackson, but the scandal was essentially fake.

Sources: Indelible Ink: The Trials of John Peter Zenger and the Birth of America's Free Press - Richard Kluger

The Birth of Modern Politics Andrew Jackson, John Quincy Adams and the Election of 1828 -Lynn Hudson Parsons