Rejecting the idea of forming a third party, Sen. Alan Dixon (D-Ill.) today urged Adlai E. Stevenson III to run for governor of Illinois as a regular Democrat by repudiating far-right candidates nominated to the party slate and promising to eliminate the lieutenant governor's office if elected.

Dixon, the highest vote-getter in state history who is seeking a second term in the November elections, opposed a possible bolt by Stevenson from Democratic ranks to form a new party and lead it into the election. He called the proposal "far too complicated, far too difficult to pull off."

But Stevenson, whose second attempt to defeat incumbent Republican Gov. James R. Thompson is a shambles, declared in a televised interview tonight that if he is unable to legally "purge" the unwanted candidates backed by ultraconservative Lyndon H. LaRouche Jr., "we won't be able to ask for straight Democratic votes because we'll have two neo-Nazis on the ticket."

Stevenson was referring to Mark J. Fairchild, who won the Democratic nomination for lieutenant governor, and Janice Hart, nominated for secretary of state. The two defeated Stevenson's handpicked candidates for the two nominations in the party's Tuesday primary election. Stevenson and Dixon easily won nomination in the primary.

Dixon, in a statement released by his Washington office, said he conferred with Stevenson today and told him of his opposition to the idea of a third party candidacy. It would "not result in any Democratic victories in the fall," Dixon said.

"They should all run as Democrats, disavow the two LaRouche candidates, and require Stevenson to promise to work, upon election, to abolish the office of lieutenant governor . . . . "

Stevenson, a former U.S. senator and heir to a powerful Illinois political name, has been an aloof party maverick, feuding with leaders of the once-powerful Chicago Democratic machine and shunning such revered back-slapping partisan functions as dinners and parades.

He appears headed his own way in this situation, as well. On Wednesday night, he ruled out what Dixon today suggested, running on the same ticket as Fairchild, who under state law is paired with Stevenson on the Nov. 4 ballot although they were nominated separately.

But if he withdraws as the regular party candidate, state election law prohibits Stevenson from simply running alone as an independent. He must form a new party and field eight other statewide candidates as well as a lieutenant governor running mate he likes.

The others include nominees for the U.S. Senate, state attorney general, comptroller, secretary of state, treasurer, and three candidates for the University of Illinois board of trustees.

Unlike many states, Illinois bars candidates from running for office as the nominee of more than one party. This means that Stevenson either must convince his fellow statewide Democratic candidates to bolt the party with him and help form the new party, or he must name rival candidates to compete with Dixon, and other Democratic nominees.

John Schmidt, a Loop lawyer heading Stevenson's effort to dislodge Fairchild and Hart, said every document filed by the two with state elections officials will be scrutinized for possible irregularities. "It's possible we'll find something, though that's not very promising," Schmidt said.

Schmidt, echoing Dixon, said a third party attempt is "nothing simple. Obviously, it's a terrible situation for Stevenson, and for the Democratic Party . . . a frightening situation [that] just crept up on people."

But one local political pollster, J. Michael McKeon, based in Joliet, said today that he warned national and state party officials more than a year ago of blue-collar voters' rising interest in LaRouche candidates in areas of high crime and unemployment.

In a position paper he wrote in January 1985, which he said was circulated to Democratic leaders, McKeon observed that in "the Joliet area, the LaRouche party reelected a significant number of [Democratic] party committeemen and won the county auditor nomination over the regular Democratic candidate.

"In interviews with union households who express a willingness to vote for LaRouche party candidates, most had no idea what the party stood for, but were fed up with the way the two major parties were handling crime and unemployment."