I hear several things when I listen to this speech. 1) King has a recognizable "Southern" and/or "African-American" accent. 2) KIng is speaking in the loud, impassioned emotional manner with which some orators address a crowd. In the days before microphones this was necessary. Today, some speakers continue to use this style, while others adopt the intimate "talking to a friend" pioneered by Franklin D. Roosevelt in his radio "fireside chats." 3) King was a Baptist minister. The Baptists are a Protestant church denomination. In the U.S. there are shades of cultural and social differences in style between denominations. There are no absolutes, but the tradition in some denominations is that the minister projects a majestic and dignified style, and in others that the minister expresses strong emotion and exhorts the congregation to respond and participate. King is recognizably using the delivery of "a Baptist preacher." The vocal delivery style results in changing the pronunciations of some words. 4) In general, the U.S. elite used to be Anglophiles and imitated British speech (the so-called "Harvard accent.") The "Harvard accent" is long gone, but it wasn't completely gone in the 1960s. Even today, when U.S. speakers are trying to sound authoritative, dignified, and correct, there is a tendency to shade pronunciation just a bit toward British pronunciation.

Many people learned to speak with a Mid-Atlantic accent back in those days. I always noticed prestigious Black people spoke with a Mid-Atlantic accent. James Baldwin is one person that comes to mind who also spoke with this accent.