It's easy for us to look back at Nixon as morally lacking in hindsight. But did people consider Nixon as being a trustworthy president before Watergate? Were people shocked by Watergate or did they kind of see it coming?
In the context of trustworthiness, Nixon's integrity was questioned basically as early as his first campaign against Jerry Voorhis, when Nixon questioned why Voorhis had received endorsements from communist groups and had not rebuffed such endorsements. There is little evidence Voorhis was aware of the communist aspects (and possibly the specifics of the endorsement itself), but Nixon had found a bell to ring that people heard.
Nixon rang that bell again, much more aggressively, against Helen Douglas in the 1950 US Senate election in California. The "issue" here is that Nixon was not actually the first to make such accusations against Douglas, (the LA Daily News publisher Manchester Boddy started that), and each candidate - Nixon, Douglas, and Boddy, attacked one another for some type of left leaning communist sympathy.
Boddy labelled Douglas as the Pink Lady, linking her with communism, and renamed Nixon "Tricky Dick". The campaign was a rather mean-spirited one across the board.
So, to be brief, people (especially his critics) had been questioning Nixon's integrity since the very start of his political career. Keep in mind that Nixon had a long political career, from 1946 through 1960, and then from 1968 through 1974. This meant that every development in his personality or political outlook could be latched on to as a new ploy or scheme and criticised as yet another deception. Nixon was well aware of this - he often complained to John Foster Dulles that President Eisenhower refused to let him act as anything other than a "hatchet man" in regards to the Democrats, and that this was hurting his credibility whenever he spoke on topics that would need bipartisan support.
The point I'm making is that, no, Nixon was not really regarded as trustworthy pre-Watergate. Effective? Absolutely. Pragmatic? Without question. More of a centrist than most people would think? Yes. But not ever truly trustworthy.
Put another way, Nixon was viewed as someone you could trust to solve a problem but who you wouldn't leave alone in a room with the good silverware.
Sources:
And archival documents at the Eisenhower Library which I can list out if need be.