There was a great variation of responses by Southern newspapers. It depended on factors like readership, location, and the individual editors. The Southern Press wasn't just one thing, with one reaction.
The black press, unsurprisingly, was way ahead of everyone in covering the rising civil rights movement. Long before Brown v Board of Education, Emmett Till, and the Montgomery Bus Boycott, they were chronicling injustices and acts of resistance. While important to the black community, these papers often struggled to stay alive and went unnoticed by whites.
In the white press, editors were often against integration, but also against mob protests and violence. Here's an example of an editorial that was reprinted in Birmingham, but originally written in Charleston, calling for "orderly method of eventually restoring control of their affairs". "Their affairs" being Jim Crow. Finding a paper that was outright advociating violence is much harder
Meanwhile, James Kilpratrick of the Richmond New Leader in Virginia wrote a series of editorials attempting to resurrect "interposition". Interposition, which is the kissing cousin of nullification, posits that a state can work to prevent the federal government from carrying out laws that the state deems unconstitutional. While interposition was deemed unconstitutional, it was nevertheless popular. Popular enough to worry Roy Wilkins and Thurgood Marshall of the NAACP and get a shout out by MLK:
I have a dream that one day down in Alabama with its vicious racists, with its governor having his lips dripping with the words of interposition and nullification, one day right there in Alabama little black boys and black girls will be able to join hands with little white boys and white girls as sisters and brother
Too, there were white newsman who stood up against segregation. Perhaps the most famous was Harry Ashmore in Little Rock, who won a Pulitzer for his coverage of the Little Rock Nine. Ralph McGill of the Atlanta Constitution, who won a Pulitzer a few years after Ashmore for his editorials, was well known for his anti-segregation stances. These guys weren't at rinky-dink rags, either. Both were important men at large newspapers.
As you can see, the editorial reaction to the end of Jim Crow was varied, based largely on who was running the paper. However, the editorial content and news content are not the same thing. Pro-segregationists papers could still possibly have decent and objective reporting. Whether they did or not is out of my depth. If anyone wants to add something, I'll mail you a kiss.
Roberts, Gene, and Hank Klibanoff. The race beat: The press, the civil rights struggle, and the awakening of a nation. Vintage, 2008.