In what may have been his last act as secretary of state, Alexander M. Haig Jr. has sent the White House long-awaited recommendations to limit U.S. arms sales to Taiwan in order to settle a major dispute with the People's Republic of China.

Official sources said Haig took the action before leaving early yesterday for a long holiday weekend at the Greenbrier in White Sulphur Springs, W.Va. "The cord hasn't been officially cut yet," White House deputy press secretary Larry Speakes said of Haig's status as secretary of state, even though the nomination of George P. Shultz to succeed Haig was sent to the Senate yesterday.

The controversy over arms sales to Taiwan has brought Sino-American political and strategic relations to a standstill for about six months and has threatened a serious deterioration of the relationship. The recommendations for resolving the dispute had been on Haig's desk for about a month following White House discussions and diplomatic feelers to the Chinese.

According to an informed account, the United States, as part of the proposed settlement, would express confidence that the quantities of U.S. arms to Taiwan will decline over time. This expression, which is short of an ironclad promise, is justified in Washington's view by continued progress toward a peaceful solution between Peking and Taipei.

The relationship of the sales limitation to progress toward peace is a delicate matter in Peking, which considers its dealings with Taiwan to be an internal matter not subject to negotiation with the United States. However, Peking has been fully consulted through diplomatic channels about statements that Haig is proposing, and U.S. officials are hopeful that Peking will approve them as a settlement of the dispute with Washington.

Another question is whether President Reagan will approve Haig's recommendation, in view of Reagan's long sympathy for Taiwan and increasingly vocal pro-Taiwan sentiments from some members of Congress.

Sens. Barry Goldwater (R-Ariz.) and S.I. Hayakawa (R-Calif.) have sponsored a letter calling on Reagan to move ahead as soon as possible to authorize new sales of F5E warplanes to Taiwan, in order to keep the military production line for those aircraft open. Hayakawa's office said yesterday that the letter is signed by 40 senators and was being sent to the White House last night.

Last week, Reagan's national security adviser, William P. Clark, met with about 15 lawmakers in a meeting arranged by Sen. Strom Thurmond (R-S.C.) to assuage congressional concern about the Taiwan issue.

The Carter administration made clear, in announcing the normalization of U.S. relations with China in December, 1978, that arms sales to Taiwan would continue. China, which never agreed to this continuation but normalized its U.S. relations nonetheless, expressed increasingly strong objections in recent months to the continued sales.

Reagan decided in January to reject the proposed sale of a sophisticated FX fighter plane to Taiwan, but at the same time approved continued sales of less advanced F5E aircraft, which Taiwan has been buying for some time. Peking rejected this decision and threatened to withdraw its ambassador from Washington and take other steps unless a long-term solution of the Taiwan arms issue is found.

Negotiations, mostly between Ambassador to China Arthur W. Hummel Jr. and the Chinese Foreign Ministry, have been taking place for months. In a bid to find a solution, Vice President Bush visited Peking in early May. The Haig recommendations flowed from that trip and subsequent administration deliberations.

Meanwhile, the secretary of state-designate continued his rounds on Capitol Hill in preparation for his Senate confirmation hearings, which begin July 13.

Sen. Christopher J. Dodd (D-Conn.) told reporters that Shultz had been under the impression that "things were getting better" in El Salvador. Dodd said he replied that "things are getting worse" there and reminded Shultz that the administration must certify satisfactory progress in land and other reforms this month in order to continue military aid to that country.

Shultz met with Reagan at the White House yesterday and sat in on his daily briefing by Clark before the president left for a California vacation. Shultz left for San Francisco later in the day to wind up his business affairs before joining the administration.