On Jan. 27, flush with the election victories that returned Congress to Democratic control and mindful of the weakened Reagan presidency, House Speaker Jim Wright (Tex.) and Senate Majority Leader Robert C. Byrd (W.Va.) promised the American people that the 100th Congress would showcase the Democrats' ability to govern.

In their joint response to President Reagan's State of the Union message, they pledged cooperation with the administration, but left little doubt that they would pursue an activist agenda with or without the president's help.

"Talk is not enough," Wright said. "What counts is action." Added Byrd: "Right now, Congress wants to get on with the business of governing."

Less than five months later, the resolve of Byrd and Wright is facing a serious test as they try to break a Democratic impasse over the fiscal 1988 budget, a document that will in many ways serve as a foundation for Congress' work through the summer and fall.

Though the Democrats are no less united in opposing Reagan's overall spending policies than when the president submitted his budget in January, over the past five weeks that cohesion has been strained by disagreements over priorities.

The dispute became an open breach on Thursday, when Senate members of the budget conference committee unceremoniously rejected an offer from their House counterparts to bridge their differences over military spending, the only real issue that prevents the two houses from agreeing on a budget.

The latest House proposal would give the Defense Department $295 billion in spending authority compared with the Senate's $299 billion, a $4 billion dispute in a trillion-dollar budget.

But on a broader level, the gap illustrates the philosophical differences between the House and Senate that could jeopardize the Democrats' effort to portray their party as an able manager of the nation's business.

"We're so close but yet so far," said Rep. Jim Slattery (D-Kan.), a member of the House Budget Committee who fears that a deadlock would send "a message that it doesn't look like we can govern."

The dispute comes on an issue that had been the best demonstration of cooperation between the Democratic leaders. Byrd and Wright have held regular consultations on the budget, agreeing on a framework that includes a controversial $18 billion tax increase.

However, that overlay of leadership unity has not contained the restiveness among the rank and file. More than a week ago, Byrd and Wright endorsed a budget compromise that would have set defense spending authority at $300 billion, $1.5 billion less than the Senate voted and $11.3 billion higher than the House approved.

But House conferees angrily rejected the compromise, characterizing it as a betrayal of Democratic principles and a giveaway to the more conservative Senate that would prolong the Reagan defense buildup and jeopardize domestic programs in future years.

An outburst by Rep. George Miller (D-Calif.) captured the irony of the situation. "If the Republicans had sent this over we would have kicked their ass all over town," Miller said, "but because it's the Democrats we're supposed to roll over and play dead."

House Majority Whip Tony Coelho (D-Calif.) also portrayed the dispute as a Democratic litmus test when he derided the Senate position.

"Some people who are interested in being presidential candidates in the Democratic Party aren't willing to compromise to get a Democratic initiative," he said in an apparent reference to Sam Nunn (D-Ga.), the chairman of the Senate Armed Services Committee.

Nunn is among a handful of Senate Democrats who have insisted on a level of defense spending that would fund the military at close to what would keep pace with inflation and without whose support the Senate version of the budget would not have passed.

As it was, Senate Budget Committee Chairman Lawton Chiles (D-Fla.) was only able to usher the budget through on a 56-to-42 vote on May 7, giving him little maneuvering room in the ensuing conference negotiations with the House.

Yesterday, Chiles argued that the last House offer not only was unacceptable to conservative Democrats like Nunn, but also might cost him liberal support because it eliminated the linkage between higher defense spending and presidential acceptance of the $18 billion tax increase.

For Chiles and House Budget Committee Chairman William H. Gray III (D-Pa.), the task of mediating a compromise has been complicated by the lack of any hope of Republican support. In both the House and Senate, the GOP has decided to let the Democrats go it alone.

"It allows one group to be so much more powerful," said Chiles, referring to the conservative Democratic senators.

Republicans have seized on the budget deadlock as proof of the bankruptcy of Democratic leadership.

"Where is the budget resolution?" asked House Minority Whip Trent Lott (R-Miss.) on the floor Thursday. "I realized at the beginning of the year that the Democrats were not going to work and agree with Republicans on budget numbers, but I am become more and more amazed, not amused, that Democrats cannot even agree among themselves."

That kind of commentary is not lost on the Democratic leadership, which met again yesterday in another effort to seek a compromise. Though they reached no accord, the leaders did agree that they can ill afford a prolonged impasse.

The last thing Democrats want is to revive the description contained in "Mr. Dooley Discusses Party Politics," in which American humorist Finley Peter Dunne wrote, "Th' dimmycratic party ain't on speakin' terms with itsilf."