I read the story in the early eighties and remember it as being called “The Jackson Killers” but can’t find any reference to it. I believe it was in an anthology of sci-fi short stories. Anyone recall it?
"The Jackson Killer", a short story by Philip E. High, first published in New Worlds Science Fiction #106, May 1961, available at the Internet Archive. You may have read it in the 1977 paperback anthology The Best of British SF 2 edited by Mike Ashley.
"Bluntly you are a paid assassin?" The words were spoken by a slender, dark-haired man who had been introduced to him as David Kearsney.
"Not an assassin, sir, a government agent from the Eliminator Corps."
"A flowery title for the same thing, isn't it?" Kearsney's face was cold." You kill men."
Lassen sipped his wine." Only a certain type of man — I'm a Jackson killer."
There was a strained silence then someone laughed a little nervously. "My name's Jackson."
Lassen made a deprecating gesture. "You confuse a name with a social malaise." He looked about him. "The work of the Corps is necessary, just as the elimination of pests is necessary."
[. . . .]
An Eliminator knew he was doomed from the moment he signed the necessary papers.
There was no short-term-office in the Eliminator Corps for, after the first few killings, he was too mentally shocked to retire with his own conscience.
After a few more, he had passed the point of no return and become to believe in his own God-like immunity.
Throughout the Empire there was no task so demanding and no walk of life so conducive to paranoia. Inevitably the agent moved from latent to positive and became as those he was ordered to destroy.
The Corps, who kept a tight check on its personnel, knew when an agent's usefulness was past and he was given what appeared to be a routine assignment.