How does Bush win in 1992: He admits he screwed up by raising taxes in the 1990 budget deal, and fires his budget director.
As for what changes? I'd expect Bush would get two more picks for the Supreme Court. Who might they be? I'm think Emilio Garza (who almost got the seat Clarence Thomas was nominated for) - ending up the first Hispanic Supreme Court justice 12-13 years before Sotomayor. The second? Probably Orrin Hatch, a senior Republican on the Judiciary committee.
That puts the court a bit more to the right: Hatch, Scalia, Thomas, Rhenquist, and Garza would form a conservative bloc.
That leads to a shift in the Evans v. Romer ruling of 1996. Instead of Colorado's Amendment 2 going down 6-3, it is upheld 5-4.
Instead, the majority opinion, written by Justice Hatch, said, "Since the Constitution remains silent on homosexuality, under the Tenth Amendment, this issue is to be resolved via the democratic process, including state constitutional amendments. To overturn this would be to exercise political will, not judicial judgment."
That ruling managed to energize the left, giving Governor Mario Cuomo and Senator Al Gore and a narrow win in 1996 over the Dole/McCain ticket.
One other effect would be that the operations in Somalia go very differently: In OTL, the Clinton Administration denied the on-scene commander's request for armor and AC-130 gunships prior to the Blackhawk Down firefight. The Bush Administration not only sends the requested assets, but they also see things through after the engagement. The death of Mohammed Aidid at the hands of Delta Force and the Rangers in April, 1994, is a huge triumph for the Bush Administration.
Cuomo is far more liberal, and presumably dovish as well. He may try to push defense cuts, and if the embassy bombings and attack on a navy destroyer go down, it could lead to a perception that the Democrats have given America another wimp - just like Carter. That could open the door for a somewhat hawkish president.
The South was trending Republican, anyhow, so I still do not see Ann Richards holding on in 1994 in Texas. However. absent Gingrich as the leader, and with George H.W. Bush as a successful president, I can see Jeb winning the Florida governorship that year as well.
By 2000, the GOP may end up beating Cuomo by putting up a Bush/Cheney ticket - only it will end up being Jeb Bush and Dick Cheney - primarily because of the "wimp factor" that is attaching to the Democrats. Jeb's Florida win will be decisive, making the "butterfly ballot" a footnote in history, and leading to a less toxic political environment. Jeb replaces Sandra Day O'Connor with John Roberts in 2002, and when Rhenquist dies in 2005, he is replaced as Chief Justice by Roberts, while Ken Starr, a former Solicitor General, is chosen as a new Associate Justice to replace Roberts.