>Can't postmodernists just shift to a system that deals with contradictions
Of course they can and they do! Heck, not just postmodernists: hardcore scientists do it (as they always did) on a daily basis, acting wildly pragmatic and idiosyncratic in their practice (using classical mechanics for launching rockets and quantum mechanics for the computers that collect the data). The epistemological point is to acknowledge that we're doing it, at least...
The problem, I think, is that maybe the ways I (or some philosophers) put it often come out as a luddite re-mystification of the world, while in fact is modernity going no brakes and full force: the hipermodernity, where even the last myth (the objectivity of scientific discourse) was debunked. It's not going back to the realm of unwarranted faith, but towards an extreme skepticism that leaves us only with hyper-specific universes of discourse (including interdisciplinary studies) and their respective truth-value systems.
Is this conclusion (that the pursuing of reality ended in its obliteration, that is, not its ending but the depriving of it) shocking and grotesque? By all means, yes. But it's ostensibly the way it happened and, rather than deny it, postmodernists try to come to terms with it and exploit it the best possible way.