Post-modernism is many things and its exact meaning is subject to argument, but I think most would agree that it explicitly rejects things like formalism and realism in favor of socially constructed narratives. In that sense, what I mean by "post-modern science" is not necessarily a rejection of scientific evidence, but a prioritization where support for the favored narrative is more important than the details of scientific evidence. We have seen this for quite a while in climate science, where alarmists, when they talk among themselves, discuss how it is more important for them to support the narrative (catastrophic global warming and, tied with this, an increasing strain of anti-capitalism ala Naomi Klein) than to be true to the facts all the time. As a result, many climate scientists would argue (and have) that accurately expressing the uncertainties in their analysis or documenting counter-veiling evidence is wrong, because it dilutes the narrative.
I think this is the context in which Naomi Oreskes' recent NY Times article should be read. It is telling she uses the issue of secondhand tobacco smoke as an example, because that is one of the best examples I can think of when we let the narrative and our preferred social policy (e.g. banning smoking) to trump the actual scientific evidence. The work used to justify second hand smoke bans is some of the worst science I can think of, and this is what she is holding up as the example she wants to emulate in climate. I have had arguments on second hand smoke where I point out the weakness and in some cases the absurdity of the evidence. When cornered, defenders of bans will say, "well, its something we should do anyway." That is post-modern science -- narrative over rigid adherence to facts.
If you want post-modern science in a nutshell, think of the term "fake but accurate". It is one of the most post-modern phrases I can imagine. It means that certain data, or an analysis, or experiment was somehow wrong or corrupted or failed typical standards of scientific rigor, but was none-the-less "accurate". How can that be? Because accuracy is not defined as logical conformance to observations. It has been redefined as "consistent with the narrative." She actually argues that our standard of evidence should be reduced for things we already "know". But know do we "know" it if we have not checked the evidence? Because for Oreskes, and probably for an unfortunately large portion of modern academia, we "know" things because they are part of the narrative constructed by these self-same academic elites.