Elizabeth Dole's appointment to the Cabinet draws attention to the progress for all American women that she has been seeking as the President's designated hitter in this field. Within two or three weeks, according to the White House, there will be news of results. Among the possibilities:
* Further appointments of women to federal posts.
* Long-term plans to promote legal and economic equity for women. These have been promised by the coordinating council on women headed by Mrs. Dole.
* Steps to correct discriminatory laws in cooperation with the states. The White House's ''fifty states project'' has come up with a thick report identifying where each state stands. The effort to move from identification to action is ''in high gear,'' says Mrs. Dole. Strengthened enforcement of child support laws is one goal.
* Proposals for congressional action to prune sex bias from the federal legal code. Last fall Senator Dole, Mrs. Dole's husband, introduced the Federal Equity Act to purge the code of about a hundred discriminatory provisions. He plans to reintroduce his measure later this month. It could call for cleansing the code of even more discriminatory provisions.
Added examples should be available from the interdepartmental task force assigned to ensure that neither current nor future federal laws or regulations discriminate on the basis of sex. A new quarterly report is due before long with presumably more openness than the earlier one that was kept under wraps until a reporter blew the whistle at a presidential news conference.
A final review of federal laws is scheduled for this spring, and a final review of regulations next spring.The intention is to clean the slate and then guard against enactment of new discrimination.
In the year of the computer, a particularly symbolic detail is the updating of the Justice Department's continuing computer search of federal laws for instances of gender bias either in language or substance. Certainly no chip should be left unturned in rooting out discrimination from those federal laws that so often provide models for legislation at other levels. President Carter's task force on discrimination discovered how entrenched bias could be: In one state an agency stopped labeling job openings for men and women but started listing some on blue paper and some on pink.
The United States is far beyond that now. Isn't it? Any civil rights group can point to multiple instances of discrimination in less comic but no less stubborn forms. Unequal pay for comparable jobs is a prime example.
So, as Mrs. Dole takes the wheel as secretary of transportation, her work for women at the White House is not only a legacy but a challenge. It is a challenge not to lapse into the ''posturing'' on women's issues reportedly recommended in a confidential administration memo that has recently come to light on Capitol Hill. It is a challenge to persevere in the struggle against what she referred to in an article in these pages as ''the shortfall between society's promise of sexual equity and the often frustrating facts of American life.''
President Reagan has chosen not to address that shortfall with means supported by the Republican and Democratic Presidents immediately before him; namely, affirmative action and the Equal Rights Amendment. Assuming the White House news mentioned above arrives as scheduled, the public should soon have some clues to the effectiveness of Mr. Reagan's alternatives.