Barack Obama denounced Republicans on Tuesday for accusing him of exaggerating the potential effect of the sequester budget cuts and then panicking in the face of public outcry over air traffic delays.

Speaking at a White House press conference to mark the first 100 days of his second term, Obama claimed it was increasingly clear that he had not been "crying wolf" when he warned about the impact of the cuts earlier this year.

The president called on Congress to agree a broader deal on the budget rather than implement short-term fixes. But he denied suggestions that it was his responsibility to shift the log jam. He told reporters: "You suggest that my job is to somehow get them to behave. That's their job."

In spite of his problems getting a deal with Congress on the budget or on other issues such as gun legislation, he denied suggestions from reporters that he was already a lame duck.

"Maybe I should just pack up and go home," he joked. "Rumours of my demise might be a little exaggerated at this point," Obama said.

When the sequester cuts kicked in on 1 March after the failure of the White House and Congress to reach agreement on tax and spending, Obama warned the impact would be dire. Initially there was little impact and the president was accused of acting like Chicken Little, hysterical in his warnings that the impact would slow recovery from recession.

"The notion was somehow that we had exaggerated the effects of the sequester. Remember? 'The president's crying wolf. He's Chicken Little'," Obama said. "And then in rapid succession, suddenly White House tours: 'This is terrible. How can we let that happen?' Meat inspectors: 'We've got to fix that.' And most recently: 'What are we going to do about potential delays at airports?'"

Obama expressed confidence that would eventually get a budget deal with the Republicans and that he was hopeful he would get one too an immigration reform bill.

At the press conference, Obama said the estimated $87bn cuts were hitting economic growth. "What we now know is that what I warned earlier is happening. [Sequestration has] slowed our growth. It's resulting in people being thrown out of work."

The White House and Democratic members of Congress initially said they would not do piecemeal deals and would only consider discussion with the Republicans on an overarching solution. But Obama, in the face of mounting air traffic cancellations and delays, caved in last week over a bill that eased restrictions on the Federal Aviation Administration spent its money. The change allowed the FAA to avoid ordering furloughs for air control staff and close air control towers.

Although Obama said he had decided not to veto the bill, that did not mean he agreed with it. The bill merely allowed the FAA to move money from long-term repair and renewal projects to paying staff in the short term. "That's not a solution," Obama said. "What we've said is that in order to avoid delays this summer we're going to ensure delays for the next decade," Obama said.

Although he described Congress as "pretty dysfunctional", he said the FAA bill at least showed a compromise could be reached and that the White House is still reaching out to the Republicans about a broader budget deal.

Paul Begala, a former Clinton adviser, tweeted: "Seems to me Potus is justifiably annoyed with the juvenile notion that he can wave a magic wand and move Tea Party Repubs to compromise." © Guardian News and Media 2013