I am reading The Uncensored War by Daniel Hallin for a research essay. He points out that in the early 60s, reporters stuck to "just the facts," focusing on current events coming from the Johnson Administration rather than analyzing, criticizing, or speculating. In the late 60s however, networks and newspapers became openly critical. Hallin says it wasn't until the late 60s that the modern op-ed page came about, and it was directly the result of the Vietnam-Watergate era. So, what changed?
There was a common thesis that the US news media transformed from a passive institution to an institution of opposition to political authority over the course of American involvement in the Vietnam War. Michael Robinson called the news media "the third party which never need faces the sobering experience of governing."
There was, as you said, a shift in Vietnam coverage by the news media from favourable and in support of the administration's Vietnam policies before the Tet Offensive in 1968, to considerably less favourable coverage after Tet. Despite this, the news media did not become sympathetic to the opposition: the anti-war movement.
So, while the idea that the media shifted to an oppositional role during the war is not supported, it did turn to more critical coverage. This is because as the war became less favourable in public opinion, the news media shifted with this change. This moved journalists away from reporting what they had been told by government officials, and naturally caused the media to become more critical, but not strictly oppositional, of the war in Vietnam.
Journalistic objectivity (insofar as not being overtly politically biased; absolute objectivism is essentially impossible) is held in very high esteem and is and always has been seen as professional in the US. So while news media became more critical, it did not lose its general ideology of objectivity.
The media in America, as concluded by Daniel Hallin, reflects the prevailing pattern of political debate. So when consensus is strong (popular support for the war), the media tends to stay within the limits of political discussion that the consensus defines. When consensus begins to break down, coverage becomes increasingly critical and diverse in the viewpoints it represents.
I hope this adequately answers your question. If you would like more information, please refer to the source below.
Source: Daniel C. Hallin, 'The Media, the War in Vietnam, and Political Support' The Journal of Politics, 46, 1 (1984) pp.2-24
If you have JSTOR access: www.jstor.org/stable/2130432