AFTER ALL THOSE decades of struggling for a taste of local self-government, Washington is now getting a bellyful of politics from City Hall - beginning with some lively jockeying for positions in this fall's big elections. More and more, we're sure to hear and see news reports with phrases such as "Mayor Washington, in what was seen as a political maneuver, today said (called upon) (ordered) (promoted) . . ." Or perhaps, "City Council Chairman Sterling Tucker widened the political gap between himself and the mayor today by . . ." And there will be more and more public consternation about the way political considerations seem to be affecting decisions in government.

As early as last August, in fact, there were raised eyebrows when it was reported that rivals of Mayor Washington had been trying for months to get the mayor out of City Hall and into an American embassy somewhere, anywhere. The group was led by Del. Walter E. Fauntroy and local Democratic Party Chairman Robert Washington, the reports went on to note. These two gentlemen just happen to be "political allies" of Mr. Tucker, "who is considered a principal threat to the mayor if the mayor should seek re-election . . ."

Mr. Fauntroy "insisted," of course, that this effort was not "politically motivated," even though it would have made Mr. Tuckeer the temporary head of City Hall. "Those involved say theirs is a compassionate campaign of civic concern that is best not only for Washington the city, but also best for Washington the man," was the way the story went. Sure. And that's why the mayor was "outraged" by the effort, according to an aide, who said, "The mayor sees it as an effort by a few politicians to hand-pick the next mayor through a backdoor maneuver and more or less take it out of the hands of the electorate."

Then there were the reports about how Mr. Fauntroy - well, supporters of his, anyway - was still trying to sell Marion Barry on the idea of running for city council chairman instead of mayor, with great descriptions of "an unbeatable team" of Fauntroy-Tucker-Barry. This way, you see, people could avoid a "devastating split" in the Democratic Party ranks. True, Mr. Fauntroy was about to endorse Mr. Tucker for mayor, but the argument given for Mr. Barry's running for chairman was that this would establish "a precedent of political tradition in the party," that Mr. Tucker, after all, is older, more experienced and has "served his time as chairman."

Now, you'd think we had some sort of scandal here. "A blatant power play," wailed the local chapter of Americans for Democratic Action, an example of "politics of a limited choice by a few political brokers who take it upon themselves to decide who has served enough time to run for higher office . . . unhealthy." Fortunately, Mr. Barry recognized a chance to have some fun with it all: "I reject his notion that order and age and time are the same as ability. If that's the case we ought to continue with Walter Washington. He's been in there. He's the present mayor and he's paid his dues."

This is politics as usual, no more and no less. And it's bound to get rougher as the season progresses.It always does, just about anywhere. the only difference in the case of the District is that some of those who go through the same sort of rough-and-tumble political process to win election in other jurisdictions - those elected to Congress, for example - will no doubt point to the skirmishes here as evidence that "these people aren't really ready" for full representation in Congress. They should know better.