When Sen. Adlai E. Stevenson III (D-Ill.) got fed up with the Senate, with President Carter, with politics in general and decided not to run for re-election, he probably did not realize the Republicans would win Illinois.

To an out-of-state observer, it may appear the Democrats have the advantage--Alan J. Dixon, who holds the coveted office of secretary of state, is running 20 per cent ahead of his Republican challenger, Lt. Gov. David O'Neal. But if Dixon wins the Senate seat, Gov. James R. Thompson will name his replacement--and the new secretary of state won't be a Democrat.

Indeed, a lot of Democratic party members probably would have preferred that Dixon not run for the Senate--a Washington politician doesn't have that much clout. Not only are Democrats likely to lose the jobs under the secretary of state's office (despite the merit system), they are probably losing one of the few Democrats popular enough statewide to challenge Thompson in 1982. With the party itself weakened by internal squabbling in Chicago between Mayor Jane Byrne and State Sen. Richard Daley, Dixon's timing seems less that stellar.

Each candidate has cultivated his party's traditional constituency. Dixon, a career politician, has received large contributions from state employees and from labor organizations; O'Neal from the Republican National Committee and Chicago businesses. O'Neal is also endorsed by several right-to-life groups and by the STOP-ERA interests, headed by Phyllis Schlafly of Alton.

But they have one constituency in common--downstate voters. Both of them are from Belleville, a city of almost 50,000 just east of St. Louis. As a result, each of them is paying close attention to the traditionally neglected interests of the moderate-to-conservative downstate farmers and laborers. Downstate residents who used to vote for the candidate who paid the most attention to them must now find different criteria.

They share several other traits, too. Both were wunderkind; Dixon was elected to the state legislature in his early 20s, and O'Neal began his politicking by winning the race for sheriff in a heavily Democratic county. Both favor free markets for farm produce and more mining of soft Illinois coal. Both call American foreign policy "indecisive."

Squabbles over campaign fund sources and candidate character have occupied a large part of this election. The Chicago Tribune last month disclosed that O'Neal charged the taxpayers for more than $62,000 worth of flights on state airplanes that his campaign should have covered. a supervisor working for Dixon admitted shaking down drivers' license applicants for contributions.

Asked in the fourth debate whether the tone of his campaign was negative, O'Neal said he had tried to be positive. "If I wanted to resort to innuendo and smear," he noted," I'd be bringing up a drunk driving charge and how a Wendy's franchise was transferred in the secretary of state's office." He was referring to traffic charges brought against Dixon in California in 1977 and Dixon's management of a hamburger franchise regulated by his office.

No matter who takes the election, however, three groups will win: moderate-to-conservative voters, downstate residents, and Republicans