The nation's governors said yesterday they will do their bit to help balance the federal budget, but they warned the Reagan administration that they will "vigorously oppose any attempts to shift costs" from federal to state and local taxpayers.

With only two dissenting votes but many expressed misgivings on the part of Democrats, the closing session of the National Governors Assication's midwater conference endorsed a resolution saying the states "are prepared to accept budget cuts" so long as they are given time to adjust and greater flexibility in the use of remaining funds.

But even with added language from the Democratic caucus asserting that the budget cuts "must not result in imposing an unfair and disproportionate burden upon the poor, the disadvantaged and the handicapped," Govs. Joseph Brennen of Maine and William O'Neill of Connecticut dissented.

Noting that there are still more than $6 billion of unspecified budget cuts to be detailed in March, Brennan said, "I'm not about to sign a black check." Many other governor were privately pursuing their own negotiations with the White House to save pet programs. Nevertheless, the bipartisan resolution supported President Reagan's view that "firm short-range measures are required to put the economy back on the track."

For the second time in two days, however, the governors reiterated that for the long term, they favored a full federal takeover of welfare and Medicaid costs as part of a tradeoff in which the states would assume primary responsibility for education, law enforcement and transportation.

At their meeting Monday with Reagan, the president made it clear he was not budging from his own desire eventually to turn welfare back to the states. He budget for fiscal 1982 would require the states to adjust to a 5 percent cap on the growth in federal support of Medicaid.

Reagan's successor in California, Democrat Edmund G. (Jerry) Brown Jr., was one of the governors who warned against shifting tax burdens under the guise of achieving economies. He said Reagan had a tendency to do that in California.

Brown and Gov. Brendan T. Byrne of New Jersey said former president Carter cautioned of the same danger in a conference telephone call he made from Georgia to the Democratic governor's caucus.

Carter, who has been publicly silent on his successor's program so far, was quoted by the govrnors, as seeing "difficulties" in some of the Reagan proposals and saying he would speak out later.

But for most, most of the governors were talking cooperation with the Reagan budget cuts -- not confrontation. Under the leadership of Georgia Gov. George D. Busbee (D), the association chairman, the strategy is to trade acceptance of most of the cuts for vigorous White House backing of much more gubernatorial discretion in use of the remaining money.

While most of the overt criticism of Reagan's economic recovery programs came from Democrats, GOP governors privately acknowledged concerns of their own with secific features. But, instead of waging a public campaign, Republican governors were seeking changes in a round of meetings with administration officials.

The term used around the conference for this was "fine tuning."

Gov. James R. Thompson (R-Ill.) was fuming about "brave talk" but artful caveats of the Democrats. But when Thompson, who has many dairy farmers in his state, was asked how he felt about proposed reductions of milk price support, he cleared his throat, observed that dairymen would be meeting soon with Agriculture Secretary John Block and said he believed something could be worked out.

Probably few governors were more enthusiastic in their public statements abut the Reagan plan than Gov. Richard Thornburgh (R-Pa.). But Thornburgh voiced concerns about the impact for Pennsylvania if Amtrak and Conrail are cut as proposed.

That is the one proposed cut, he said, that gave him a "real dig-in-my-heels propensity." Thornburgh has been privately presenting his case for federal railroad aid to Transportation Secretary Drew Lewis.

Lewis has also been talking to the folks from Michigan. Gov. William Miliken (R) is trying to salvage funds for a Detroit subway and bus service that had originally been promised in the Ford administration.

"We need that desperately and we've had commitments from two administration and we're trying to keep it alive," Milliken said.

Milliken and other Midwest govenors hard-hit by the recession in the auto industry are to meet wit Reagan on March 6, four days before his final busget proposals go to Capitol Hill. They want, among other things, some changes in the president's tax proposals to provide incentives to modernize old plants and factories in their states.

Vermont Gov. Richard Snelling (R) told reporters that he supports the president's programs without equivocation. He also talked about the need for some fine tuning but declined to be specific.

Later, he acknowledged that he has been designated by other governors to go to the White House to plead for salvaging some features of the Appalachian Regional Commission and other similiar regional commissions across the nation that the administration has proposed killing.

How much would that fine tuning cost?

"Twenty-five million dollars," Snelling responded tersely.