"Competence, not controversy." That's Henry E. Hudson's campaign slogan as he battles to unseat the incumbent Arlington County prosecutor in one of Northern Virginia's hottest political races this fall.

"If [Hudson] can't handle controversy, he shouldn't be running for this job," replies Arlington's combative incumbent prosecutor, William S. Burroughs jr.

A long-running dispute stemming from a complex double-murder case has beset the Arlington prosecutor's office, providing a ready-made campaign issue for Burroughs' political opponents and focusing unusual attention on the prosecutor's race in the Nov. 6 election.

The Arlington prosecutor's office is not alone. Controversy also surrounds the campaign for prosecutor in neighboring Alexandria, where former prosecutor William L Cowhig resigned amid a bingo scandal.

John E. Kloch, a Democrat appointed to succeed Cowhig last February, is seeking to hold off challenges by Republican Barry R. Poretz and indepentent John E. Kennehan in a special election designed to fill the remainder of Cowhig's term. The cloud left by Cowhig over the prosecutor's office remains a central issue in the campaign.

Although these controversies have dominated much of the Arlington and Alexandria campaigns for commonwealth's attorney -- as the prosecutor's office is known -- it is unclear whether they will be decisive factors in either election contest.

The turnout in Arlington and Alexandria is expected to be light. The candidates are spending only modest sums on their campaigns. Few other political offices are at stake in the election. And some veteran political observers say both races are likely to be determined mainly by nuts-and-bolts aspects of partisan politicking, including the strength of local campaign organizations, party loyalty among voters and whether voters recognize candidates' names.

Although both Arlington and Alexandria traditionally have been viewed as predominantly Democratic, Republicans have made significant inroads there in local elections in recent years. Some politicians say they expect the prosecutors' races to be close.

Burroughs, 41, a Democrat who has been Arlington's prosecutor since 1974, has repeatedly defended himself during the campaign against charges stemming from the double-murder case raised by Hudson, an independent running with Republican backing. Hudson, 32, was formerly Burroughs' chief assistant.

Burroughs narrowly withstood a primary election challenge by Arlington County Board member John W. Purdy, who also criticized the prosecutor's handling of the murder case. Burroughs defeated Purdy, 2,506 to 2,280. But the June primary ended with such bitterness that Purdy remains openly critical of Burroughs and says he has not endorsed his reelection.

At the center of the controversy is the 1977 slaying of Arlington real estate agent Alan Foreman and his fiancee, Donna Shoemaker. A former real estate agent, Richard Lee Earman, was initially acquitted of murder, but later agreed to plead guilty to conspiring to murder the couple. Joseph N. Martin, a former insurance salesman, currently is on trial charged with hiring Earman to kill Foreman and Shoemaker.

The murder investigation led to an initial rift between Burroughs and the Arlington police department, a state police probe of the way Burroughs handled the murder investigation, and a continuing dispute between Burroughs and Virginia Attorney General J. Marshall Coleman, a Republican.

Burroughs describes the state police inquiry as unwarranted. Coleman has said it was justified because of allegations indicating that Burroughs might have accepted bribes and engaged in other improper conduct during the murder investigation. The state police probe, nevertheless, exonerated Buroughs of any criminal wrongdoing.

In the election campaign, Hudson says the central issue is a lack of "public confidence" in the county prosecutor. To underscore his charge, Hudson points both to Earman's first trial, which resulted in an acquittal, and to Burroughs' battered relations with other law enforcement agencies.

Hudson contends that Burroughs mishandled Earman's trial by failing to prepare properly for the court proceedings, changing the prosecution's argument in the middle of the trial and failing to make use of a confession given by Earman.

Burroughs disputes all these assertions. He argues that Earman's confession could not legally have been admitted as evidence. He says he would have preferred not to alter his argument to the jury, but had no choice because new evidence came to light during the trial. He rejected Hudson's criticism of his preparation for the trial, saying, "He's just wrong."

Despite the continuing controversies, Burroughs declared in an interview, "We have an excellent record in working with other [law enforcement] agencies." He complained of repeatedly inaccurate news reports about his office in the press. Burroughs has sued both The Washington Post and The Washington Star for libel, seeking $1 million in damages from each paper.

Burroughs also contends that emphasis on the controversial murder case has distorted the election campaign by sidestepping other major issues. These, he said, include his office's high rate of obtaining convictions, his recruitment of women for jobs as assistant prosecutors, his efforts to rid Arlington of massage parlors and his support of handgun control legislation.

In Alexandria, Cowhig's legacy has cast a shadow over the prosecutor's race. Cowhig resigned Feb. 23 after he was acquitted in two trials of bribery and gambling charges. A third gambling indictment against him was dropped.

Kennahan, 55, a former Alexandria commonwealth's attorney, has bluntly raised the issue, charging that "a heavy cloud of suspicion" hangs over both his rivals for the remainder of Cowhig's term, which ends in 1981. Kennahan has attacked Kloch as Cowhig's "hand-picked deputy" and sought to line Poretz to Alexandria's bingo scandal by noting that Poretz's law firm once represented a client with ties to allegedly illegal bingo activity.

Poretz, 36, has raised the issue only indirectly, by campaigning on a "new direction" for the prosecutor's office. "The mood of the community is one of no faith, trust or confidence in the office of commonwealth's attorney," he says. He would change the office, Poretz says, by a policy of initiating his own investigations.

Kloch, 38, who had been the deputy Alexandria prosecutor since 1974 before being appointed to succeed Cowhig, defends himself against the political charges by asserting that he was unaware of the allegations involving Cowhig until they surfaced publicly. "There was no way I could have known," Kloch says. He has the endorsement, Kloch notes, of Alexandria's former special bingo prosecutor, Edward J. White.

In a political talk last week, Kloch said he is running as "a professional prosecutor" whose integrity is not in doubt. "I was Bill Cowhig's deputy -- that is a historical fact," Kloch said. "I will tell you candidly that if I had to do it again, I would do it exactly as I did."