nder strong White House pressure, the nation's governors today backed away from a go-it-alone stance on federalism and agreed to try again for agreement with President Reagan on the terms for shifting many federal programs to the states.

Reversing its executive committee stance, adopted only two days ago, the National Governors Association in its closing session voted to present the organization's views "to the president to seek his approval," before taking them to Congress next year.

In a separate action, the governors voted, 26 to 10, in favor of a simpler version of the Senate-passed balanced-budget amendment. The vote was a clear show of sentiment on the issue, and barely failed, by the switch of one vote, of the three-quarters majority needed for adoption as official policy.

The spirited debate on the bipartisan resolution had critics like West Virginia Gov. John D. (Jay) Rockefeller IV (D) warning that its passage would "wreak devastation on the states," while such supporters as Wisconsin Gov. Lee S. Dreyfus (R) said that failure to impose spending discipline on Congress would bring the country to "economic suicide."

Virginia Gov. Charles S. Robb (D) voted with the bipartisan majority for the resolution while nine Democrats and Vermont Gov. Richard A. Snelling (R) provided the negative votes that blocked the amendment from becoming official policy.

As drafted by Arizona Gov. Bruce Babbitt (D) and Tennessee Gov. Lamar Alexander (R), the resolution criticized the Senate amendment as "needlessly complex" and suggested a simple provision saying that, except in time of war, "Congress shall pass no law, except when three-fifths of the whole number of both houses shall deem it necessary, resulting in total outlays in any fiscal year in excess of total receipts."

Snelling, the outgoing chairman of the NGA, was the main target and victim of a White House power play that turned the governors around on the basic strategy for the federalism initiative. Snelling has led the governors' negotiations with the White House ever since Reagan announced the outlines of the plan last January.

On Sunday, he told the executive committee that while substantial progress had been made in narrowing the differences with the White House, remaining disagreements on the future of welfare and Medicaid programs and the long-term financing of turnback programs were such that "it no longer seems prudent to pin our hopes for a new federalism on the outcome of any negotiations with the White House."

Over the objections of some Republicans, he pushed through the executive committee a proposed "action plan" that had its main focus on "working with Congress toward enactment of a federalism reform initiative that incorporates the historical principles of the NGA."

He was backed Sunday by the new NGA chairman, Utah Gov. Scott Matheson (D), and together they were able to brush aside objections to the "bypassing" of the White House.

But presidential assistant Richard S. Williamson warned immediately that "the bottom line" the governors had to face was that "there will be no federalism initiative unless Ronald Reagan supports it."

Yesterday Williamson and other administration officials here warned governors individually that Reagan was angered by newspaper accounts of the executive committee action. Late in the day, Williamson had a meeting with Snelling, Matheson and two Republican governors with close ties to Reagan -- Nevada Gov. Robert List and Nebraska Gov. Charles Thone.

List argued that an effort by the governors to strike their own deal with Congress would bring "a premature death" to the federalism plan. Thone, a former member of the House, said that "unless we get a strong thrust from the president, it will never fly" in Congress.

According to one of those present, Williamson's basic message was that "You guys don't have anybody else but Reagan to take you to this stance, and if you don't show more flexibility, we may just tell you to go to hell."

The result was a substantially rewritten "action plan," whose first plank is to "continue discussion with the president regarding his federalism initiative." It only fleetingly mentions consultation with congressional leaders, along with state legislators and local officials.

Williamson expressed his pleasure at gaining a "good conciliatory resolution" and said there is "room for movement" from the administration on the remaining points of difference with the governors.

Matheson said the negotiations would go better if Reagan is "directly involved," as he promised some governors in phone calls Friday he would be, and if "someone other than Dave Stockman," the director of the Office of Management and Budget, is negotiating for the White House. But Matheson also told reporters that the governors still have basic "philosophical differences" with Reagan's opposition to the federalization of all cash-welfare programs, and he emphasized their tough bargaining position by naming Snelling as the "lead governor" in further talks with the White House.

He also held open the option of a "second track" in which the governors would develop their proposal for consideration by Congress.

Matheson said he wanted to broaden the talks with Reagan beyond the issues of "swaps and turnbacks" to include the ongoing reductions by Congress in federal-aid programs. But he noted that with today's vote on the balanced-budget amendment, where he was in the minority, the governors are in the position of saying that reducing deficits is "the highest priority."

By contrast to the sharp debate on the balanced-budget amendment, a resolution reaffirming support for a new try for an Equal Rights Amendment went through without debate.