Texas Democrats, eager for a November clash with incumbent Republican Gov. Bill Clements, faced another month of internal strife today after Attorney General Mark White ran a strong race in the gubernatorial primary but failed to win the Democratic nomination outright.

White, a conservative Democrat, captured about 45 percent of the vote in Saturday's primary and will face railroad commissioner Buddy Temple, who got about 30 percent, in a June 5 runoff.

In congressional races, Democratic Reps. Phil Gramm and Jack Brooks made it look easy.

Gramm swept more than 60 percent of the vote to defeat John Olin Teague, the son of former congressman Olin (Tiger) Teague, and two other challengers in the 6th District between Dallas and Houston.

Brooks, whose poor showing in 1980 sparked a concerted challenge from four conservatives, mounted an impressive campaign to win renomination without a runoff in the 9th District in southeast Texas. Brooks got 53 percent of the vote, to 22 percent for Bubba Pate, his nearest competitor and 1980 nemesis.

Gramm had hoped for a big vote to prove to Democratic critics that his constituents approved of the leadership role he played in helping President Reagan and the Republicans put through their budget and tax program last year. He used his victory statement to send a message.

"These totals tonight say in very clear terms to the people in our district and to the people who are concerned in the whole nation that the American people have staying power, that they never thought we were going to reverse 15 years of irresponsible spending and taxing in six months," he said.

The underfinanced Teague had attacked Reaganomics and Gramm's desertion from the Democrats, but his campaign never caught fire.

Brooks, who lost his home county to Pate in 1980, stormed back on Saturday with better than 50 percent around Beaumont, which proved the key to victory.

Brooks ran as a Democrat still committed to the social programs of the Great Society in the face of a conservative attack, and he referred to that record Saturday after it was clear he had survived.

"My 30-year record in Congress reveals an unwavering commitment to these beliefs and I pledge to continue working for all the people in this district with the same determination and energy that I first brought to the office in 1952," he said.

In the U.S. Senate primary, Rep. Jim Collins of Dallas won the Republican nomination with about 53 percent of the vote. He will face incumbent Sen. Lloyd Bentsen.

Voting was extremely light Saturday, as a series of lackluster races and a long squabble over redistricting caused most voters to skip the primary. A computer malfunction slowed final results in most races.

In the state's liveliest race, for state agriculture commissioner, former Texas Observer editor Jim Hightower swamped incumbent Reagan V. Brown.

Hightower, who ran a populist campaign and sought to bring together family farmers and consumers into a working coalition, got about 60 percent of the vote.

Brown was hurt in the final week when he referred to Booker T. Washington as the "great, black nigger," in what appeared to be a disastrous slip of the tongue.

Republicans are now looking for a strong conservative to run against the liberal Hightower.

Four years ago, Texas Democrats lost the governor's chair to Clements and they believe that his abrasive style and the national recession make him vulnerable in November.

After his strong showing on Saturday, White enters the runoff as the odds-on favorite to defeat Temple, a former state representative and son of Arthur Temple, vice chairman of Time Inc.

White and Clements have been adversaries since they came into office. Both are strong-willed men with short tempers, and they have clashed on a variety of issues, most notably White's handling of the federal lawsuit against the Texas prisons.

White believes only a conservative Democrat can beat Clements. He was elected attorney general in 1978 by defeating James A. Baker III, now White House chief of staff.

In other contested congressional races, two conservative Republicans will meet in a runoff in the heavily Republican North Dallas 3rd District that Collins vacated to run for the Senate. The two are Kay Bailey Hutchinson, a former state representative and vice chairman of the National Transportation Safety Board, and Steve Bartlett, a former Dallas councilman.

Democratic Rep. Jim Mattox of Dallas quit his 5th District seat to run for state attorney general. State Rep. John Bryant easily won the Democratic nomination over a conservative opponent and is favored to win in November. Mattox led in his attorney general's race but did not gain a majority and faces a runoff with John Hannah, a former U.S. attorney from East Texas who is considered a tough opponent.

In the 14th District near Corpus Christi, former congressman Joe Wyatt began an attempted comeback by winning the Republican nomination. He quit the House in 1980 because of a drinking problem.