A few random thoughts. Start off by The President's Men smearing Woodward and Bernstein, to the extent that investigative journalism is severely hampered. Effectively, libel laws get introduced that make the defence through truth almost impossible to meet. That effectively hinders the Free Press protections. The financial problems that this causes the Washington Post are sufficient to enable our good friend Rupert Murdoch to buy it up, and the politicisation of the media takes off.
As a side effect, we can introduce a dose of anti-Semitism through Bernstein: "Jews controlling the media" type tropes becoming more widespread, and how the media was un-American, and by extension ...
Vietnam was still going on, just about. The Post was one of those that were not in support of the war. So we can string US involvement out for maybe another couple of months.
The attitudes towards draft dodgers will be rather more severe. No Watergate almost certainly means no Carter and no pardon for draft dodgers. However, there was an economic crisis, and the Vietnam Veterans will still face their problems with pretty minimal support, so the distressed war veteran will still be a thing, possibly more so.
Nixon remaining in place, with unanswered questions splits the country further. A damaged Nixon in place and Kissinger as Secretary of State basically means that relations with Britain goes into the toilet, particularly with regard to Northern Ireland. We'd certainly get a more unfortunate situation in Cyprus in 1974.