So I mean, postmodernism is a pretty "diverse" field, and while there are some who would claim to be full relativists, they aren't usually taken very seriously.
What someone like Herbert Marcuse would claim is that there is something like a "definite negation," or Jacques Derrida would call a "trace;" that is, there, sitting behind every signifier (of de Saussure's structural linguistics) its negation of another signifier that defines its position in an ordered chain or a symbolic network. They would not argue that things are relative; rather, social life is regulated by these vast signifying networks that one becomes "subjectivated" in (begins to understand oneself as a subject), where one's "agency" also stands at the limits of this intelligibility: as they say, everything you "decide" is already pre-conscious; every notion of what is "possible," is limited by what is already "intelligible," i.e. what operates in the social-symbolic function as it, like an algorithm, constantly moves and exchanges its various symbolic orderings in what we call the "economy." To be a true agent means to do something which radically restructures this network and radically redefines all its terms: Foucault's "fearless speech," Badiou's "Event." The latter thinker actually explicitly rejects relativism (and is in some ways a platonist--actually many so-called "postmodern" thinkers, Whitehead among them, are considered platonists.)
The position you claim is from "post-modernism":
>"claims of truth are just assertions of power"
finds its origins in Nietzsche, who famously claimed that truth, morality, beauty, etc. were invented by the weak because they feared the powerful and they were jealous of the fact that they couldn't kill their enemies, eat fully, and have sex all the time (Master Morality, which was just doing whatever you wanted). So they rose up, took over, and invented these categories and systems of valuation in order to justify a new regime of power where nobody was able to do what they wanted, and everything was regulated by vast systems of control justified by these metaphysical notions (he refers to this as "Christian Morality.") He claims that Christian morality, after centuries of struggle, has finally won out in the Bourgeois revolutions of the 18th and 19th centuries, but now "God" has been subordinated to "Science" while retaining all the metaphysical baggage of christianity "truth, beauty, morality," etc. This is from Nietzsche's Genealogy of Morals, the argument from which was expanded in Foucault's Discipline and Punishment (the most widely read book from what most people would call "postmodernism").
>truth that is actually true, true in the old sense of corresponding to the reality of what is
The so-called "correspondence theory of truth" is considered a fallacy of the layman in academic philosophy. Truth has always been, in logic, a category: a value, specifically a boolean value. One can make claims about the structure and order of logic and the relation of that to psychological, social-processes (as Frege, Husserl, and many other "anti-psychologists" of the 19th and 20th centuries did); but to claim that language, a signifying act, somehow directly corresponds to the physical objects that it categorizes, can't really be valid: as Shakespeare said "a rose, by any other name, would smell as sweet." There is no reason why the words we use have any correspondence to experience--even if there is surely some reason why words can evoke experiences, emotions, like when one hears the name of a past-lover. But that can be attributed to what the "postmodernists" claim: the name evokes an entire symbolic structure and brings it to the surface of consciousness.