Martin Luther King Jr. almost certainly had affairs. His friend and Southern Christian Leadership Conference successor, Ralph Abernathy, wrote an autobiography titled And The Walls Came Tumbling Down in 1989, in which he discusses King's particular struggle with women on the road.

Much has been written in recent years about my friend's weakness for women. Had others not dealt with the matter in such detail, I might have avoided any commentary. Unfortunately, some of these commentators have told only the bare facts without suggesting the reasons why Martin might have indulged in such behavior. They have also left a false impression about the range of his activities.

Martin and I were away more often than we were at home; and while this was no excuse for extramarital relations, it was a reason. Some men are better able to bear such deprivations than others, though all of us in SCLC headquarters had our weak moments. We all understood and believed in the biblical prohibition against sex outside of marriage. It was just that he had a particularly difficult time with that temptation.

...

A recent biography has suggested without quite saying so that Martin had affairs with white women as well as black. Such a suggestion is without foundation. I can say with the greatest confidence that he was never attracted to white women and had nothing to do with them, despite the opportunities that may have presented themselves.

The book doesn't discuss anything about prostitution or orgies. These claims are more often attributed to J. Edgar Hoover's FBI, which considered MLK a serious threat to America due in part to the SCLC's ties to Socialist/Communist organizations. They engaged in a propaganda and smear campaign that involved constant surveillance, including bugged hotel rooms which allegedly captured his affairs on camera (famously sent to King's wife by the FBI itself), and outright fabrication when they couldn't find the evidence needed to really oust King.