I'm a strong believer that 1976 was much less of a poisoned chalice than people tend to think. While there were certainly economic difficulties, Carter's loss in 1980 is in large part attributable to preventable bungling of administrative management and foreign affairs. While voter fatigue and a drooping economy would drag down the Republicans' chances, I think it's likely that they win in 1976 and that it would be possible for them to win in 1980 without Watergate.
The conventional wisdom would be that Richard Nixon's favourite cabinet member, John Connally, would succeed him as the Republican nominee. While Connally was most definitely a conservative, it wasn't his brand nearly as much as it was Ronald Reagan's. If Connally were to win the nomination, he would have to take up all the space in the party's centre-right to lock out the likes of Bob Dole, then convince enough moderates that might otherwise support the likes of Charles Percy or Howard Baker that he's the only viable alternative to Reagan. Likewise, Nelson Rockefeller is very likely to run for a final doomed effort at the nomination. That would put Connally in the exact same position Nixon was in eight years earlier: as a desirable middle point between Reagan and Rockefeller. All this to say, there would be lots of ways for his candidacy to be derailed, but he would be the perceived frontrunner going into the primaries.
As for the Democrats, there would be much less of a mood for an outsider populist without Watergate. This would weaken Jimmy Carter and strengthen the more establishment candidates. According to polling IOTL, Hubert Humphrey was the preferred candidate of most Democrats. Despite that, Humphrey was very uncomfortable with the primary process and frequently stated that he would not run in the primaries, but would accept a draft movement or the nomination at a brokered convention. Because of his grasp of the still-new primary system I think Carter would still be a formidable candidate, but if the Anyone But Carter establishment candidates do well enough to prevent him from winning before the convention, and if there isn't a clear frontrunner among their number, then Humphrey would be the go-to choice for the nomination. Humphrey would probably feel obligated to ask Carter to be his running mate. Other than Carter, Humphrey was also considering the idea of a unity ticket with George McGovern as his running mate. In the event that they win, McGovern would become president in 1978 upon Humphrey's death!
In a Connally v. Humphrey match-up without Watergate, I think Connally would win but that Humphrey would do a lot to rebuild the Democrats' strength in the Midwest and Mid-Atlantic. Not super close but not a blowout either. As a side note, Eugene McCarthy would not have run his independent presidential campaign in 1976 if Humphrey was the Democratic nominee; he had significantly mellowed on Humphrey after their rivalry in 1968 and considered him well-equipped to be president. IOTL McCarthy got about 1% of the vote nationwide, mostly at Carter's expense, and without him on the ballot Carter almost certainly would've won in Oregon.
Given his history of corruption I think that a Connally Administration would be a sinking ship by the end of his term. If an investigation into Connally is thorough enough it might even uncover Watergate a few years later than IOTL! However, I think a more upstanding Republican with a good relationship with Congress, like Baker, could run a competent enough administration that he could be re-elected in 1980.
