Nixon gets impeached/resigns in late 1973, before Ford is nominated and confirmed as vice President.
Before or after Agnew is already in trouble though? Are you postulating Agnew somehow avoids the OTL scandal completely, or its onset is so delayed that Agnew stepping into power is not itself unthinkably problematic on all sides?
OTL I think Agnew's troubles were indirect accelerants of Nixon's Watergate crisis. Indirect in that Agnew's scandals might have been mostly under the radar long after Watergate had started to become quite regular and alarming news. I personally was in second and third grade at the time, but I distinctly remember a political editorial cartoon by I cannot remember which cartoonist showing Agnew in a very hot tub of water with Nixon, and the President (the notion he was anything less did not cross my young mind, living in a conservative military family in the Deep South, and him being the first President I had any living memory of) smirking at his VP saying "you get used to it after a while."
Still, I think it is plain that Agnew also being stained with deep scandal must have contributed to Nixon's sense of strategic weakness and the sense of his opponents that they could press on and actually oust him, especially after Agnew was gone and replaced by Ford.
It would be possible technically to meet your specs with minimum deviation from OTL by Nixon deciding if he was going down he would screw his opposition good and proper by creating maximum chaos, and resigning immediately after Agnew did without naming a replacement. Except that remember the Democrats controlled the House continually from the mid-50s under Eisenhower all the way to 1996, and the only reason he was able to get a fellow Republican to follow him was taking the time to appoint Ford first, who was House
Minority leader. Skipping his prerogative to appoint Agnew's replacement would make the House Majority leader President, and I am quite sure Nixon would have regarded that as the opposite of victory, on any terms however trollish. He'd do better to either try and protect Agnew somehow (but that would surely accelerate the spreading impression he was dangerously out of control and irresponsible) to guarantee Agnew kept the place. (But I am also sure he would bitterly resent having to elevate his attack lap-dog otherwise so plainly lacking in Presidential merit as the best way to RF his enemies left to him, if it were an option at all). Or else appoint someone else less congenial to the Democrats than Ford to replace Agnew. The trouble there is, I am not sure (we could look it up) but I believe the Amendment modifying the process of Presidential succession, the 25th if I am not mistaken, stipulates a replacement VP go through advice and consent process in the Senate, which was also Democratically controlled at this time--not to the one-sided degree later Senates would be controlled in the post-Gingrich era, but again his options are constrained by what is acceptable to his political foes. He could probably get GHW Bush though I daresay a lot of eyebrows would be raised along the lines of people saying "
Who?" But one reason he could get Bush is that he was considered a lot more moderate than he later would be--in context of the times, pretty right wing to be sure, but not Goldwater rightwing. Maybe he could get Goldwater passed by the Senate? But would Goldwater be a suitable custodian of the Nixon Legacy? Reagan I think would fall between stools completely--Nixon and he had little love lost between each other and the Democrats in the Senate would not like him either; Goldwater at least had gravitas and was one of their own.
No, the only way to get this is to free Agnew of all scandal and still have Nixon going down while Agnew stands aside with apparently clean hands. Which itself seems dubious to me--he was in fact quite guilty and so as Nixon was going down his exposure was a matter of time. His reputation was none to good before the various scandals caught up with the two of them. So we might have to go farther back and have Agnew keep his hands out of the cookie jar, but those cookie jars seem irresistable to Republicans, especially of the postmodern kind of breed Nixon was the historic flagship of--which makes logical sense to me as modern conservatism seems centered on the idea of being on the winning team being the whole point of political struggle, and winning being defined in terms of oligarchic privilege--to victors go the spoils, and wealth is its own justification. To such a mindset, to fail to cut oneself on in special deals is to forego the whole point of the struggle.
Yet another earlier POD to the POD is for Nixon to have picked someone other than Agnew in the first place back in 1968. Such a person might not be the suitable partner to The New Nixon Agnew was of course, which is precisely how and why they might stay out of trouble. It is plausible I suppose that such a different VP might stand by while Nixon's ship sinks, but clearly again Nixon would not have the gratification and vindication he would desire to salvage.
Have fun trying to figure who else he might have picked in '68 without undermining his brand and yet having someone reasonably bulletproof. Me, I figure that in addition to ideological compatibility Nixon was indulging in assassination insurance at least a little, much as the Bushes later would by picking someone so offensive to so many that would be assassins might think twice.