Democrat Michael Dukakis wraps up another busy round in his search for a vice presidential running mate, meeting Wednesday with Rep. Lee Hamilton of Indiana. And the campaign of Vice President George Bush breathed a quiet sigh of relief with the resignation of Attorney General Edwin Meese III.

Dukakis' search for a running mate occurred against the backdrop of a budget crisis at home in Massachusetts. The money crunch forced him to agree to new taxes - an issue that was quickly seized upon by the Republicans.Dukakis also came under criticism from President Reagan, who on Tuesday also gave the vice president his strongest vote of confidence yet.

Bush is "the kind of man I want to follow me," Reagan told a group of conservatives at the Old Executive Office Building. He said Bush would continue the conservative revolution begun under Reagan.

Bush was traveling to his adopted home state of Texas today for a meeting of the League of United Latin-American Citizens, the largest Hispanic civil rights organization in the country.

Dukakis on Tuesday brought in Sen. Albert Gore Jr. of Tennessee and Rep. Richard Gephardt of Missouri for face-to-face talks in Boston, and today was meeting with Hamilton. Gephardt said he'd take the job if it were offered, but Gore would not comment.

The former Dukakis rivals treated him harshly during the heat of the primary campaign but both eventually endorsed the Massachusetts governor.

Dukakis is also to travel to Arizona Friday to meet with another former rival, Bruce Babbitt.

But Bruce Garmella, who heads Dukakis' national advance team, refused to speculate whether Dukakis was considering the former Arizona governor as a running mate or whether Dukakis planned to solicit Babbitt's advice on the matter.

Jesse Jackson, who spent the July Fourth holiday with Dukakis, dismissed contenders other than himself.

"The others under consideration are unknown quantities beyond their districts and beyond their home states," Jackson said in Boston before leaving for New York, where he was holding a series of editorial board meetings today.

The private sessions with prospective running mates fueled speculation that the certain Democratic nominee was moving closer to naming a second.

While Dukakis conducted interviews at home, his closest adviser, Paul Brountas, began a four-day stay in Washington. Campaign aides said Brountas was conducting research on the vice presidential contenders and talking with party leaders.

Meese's resignation Tuesday was welcome news to the Bush campaign, which viewed the attorney general's legal problems as a liability. Although Bush never publicly criticized Meese or called for his resignation, he issued a statement saying: "Ed did the right thing and I wish him well."