BEING POLITICALLY lonesome is not a novel experience for Mrs. Ada Mills of Clarksville, Ark. Until early last year -- and for as long before that as most of her party colleagues can remember -- Mrs. Mills was the state treasurer of the Arkansas Republican Party. Since last Saturday, Mrs. Mills had been the only national convention delegate chosen anywhere who is pledged to former Texas governor John B. Connally. As of today (and for who knows how many tomorrows), Mrs. Ada Mills of Clarksville, Ark., is the ten-million-dollar delegate.

Ten million dollars. That's what Mr. Connally has raised for his now-struggling presidential campaign. The man who only weeks ago was bringing Republican audiences to their feet with his oratory, and contributions to his campaign coffers with his eloquence in the board rooms, is now in deep political trouble. The unmistakalbe signs of a faltering campaign are evident: personnel layoffs, announcements of changes in campaign leadership and strategy and lowered expectations. The election returns are just as discouraging for Connally supporters: a weak fourth in Iowa, followed by last weekend's disastrous showing in Puerto Rico, where Mr. Connally received less than 1 percent of the vote in the primary that former ambassador George Bush won handily.

Mr. Connally, to many political observers, seemed the perfect opponent for President Carter only a couple of months ago. Forceful and decisive, he gained in comparison with the incumbent, who appeared to be neither. The principal impediment to Mr. Connally's march to the White House was, it seemed, former California governor Ronald Reagan.

But even then, certain stubborn practitioners of the arcane art of polling kept insisting that their survey numbers consistently reported a large negative reaction -- generally up to one-third of the Republican electorate -- to Mr. Connally. In practical terms, that meant only two-thirds of Republican voters were even listening to him.

And voters were the problem. The former Texas governor's poll numbers and popular support did not improve in any constituency to which he was exposed. Quite the opposite: he lost points and votes. He was, after all, a Democrat more recently than Mr. Reagan and in a way that the Californian never was. He is in some sense a "statist" by conservative lights, a big-government/big-involvement man, who, moreover, had violated a special piece of conservative dogma in his wage-price control days with President Nixon. His public acquittal somehow never translated into his public exoneration. In a period in which the Iran hostage crisis made painfully evident the limitations on American power, his "Big John" style never came off as an appropriate alternative.

Then there have been flaws in the Connally campaign. First, Mr. Connally has run a "national" campaign at a time when the secret to electroal success lies obviously in running contemporaneously a few statewide campaigns. Mr. Connally exudes a certain kind of bluster and pride that is very little help to trying to recruit Iowa or New Hampshire voters, one by one, in kitchens and living rooms. In fact, it is almost impossible to imagine Mr. Connally in some ordinary kitchen chatting about inflation. The Connally approach, which failed in Arkansas and elsewhere, was to bring a couple of hundred people to a resort for a weekend of good food and recreation and charisma.

It has not worked yet, and the odds are getting longer daily that it will not work. The once-proud "national" strategy of only a few months ago has been reduced to what was called in Vietnam a modified enclave operation: trying to hang on in South Carolina and other selected southern precincts. Mr. Connally won audiences. He won financial support like nobody else in the race. But, in the hard world of presidential politics, he has yet to show that he can win votes.