There is a Wikipedia article for the word Negro, which may shed some light on the issue. Within that article is a section on the history of the use of the word in the US. It appears the usage by MLK is at the end of the time period where the word Negro was more accepted (emphasis mine):
"Negro" was accepted as normal, both as exonym and endonym, until the
late 1960s, after the later Civil Rights Movement. One well-known
example is the identification by Martin Luther King, Jr. of his own
race as "Negro" in his famous "I Have a Dream" speech of 1963.
and that it fell out of acceptable use after the Civil Rights Movement:
some black American leaders, notably Malcolm X, objected to the word
Negro because they associated it with the long history of slavery,
segregation, and discrimination that treated African Americans as
second class citizens, or worse.
Note that since this is listed as 'normal' terminology for the time, and also
both as exonym and endonym
we can infer that the word Negro was used by white politicians.
A quick search brings up a speech by LBJ, March 16, 1965.
There is no Negro problem. There is no Southern problem. There is no
Northern problem. There is only an American problem. And we are met
here tonight as Americans--not as Democrats or Republicans-we are met
here as Americans to solve that problem.
[The word Negro appears 11 times in this speech.]
Another speech, this time by JFK, on June 11, 1963 has the following:
The Negro baby born in America today, regardless of the section of the
Nation in which he is born, has about one-half as much chance of
completing a high school as a white baby born in the same place on the
same day, one-third as much chance of completing college, one-third as
much chance of becoming a professional man, twice as much chance of
becoming unemployed, about one-seventh as much chance of earning
$10,000 a year, a life expectancy which is 7 years shorter, and the
prospects of earning only half as much.
[The word Negro appears 12 times in this speech.]
So we can see that both individuals you cite as your examples used the word Negro in public address during the time frame noted. It was just another word.