I have a graduate degree in literature and, for what it's worth, even with years of training I find a lot of that stuff unintelligible. At the time it struck me as a classic emporer's-new-clothes situation. I would occasionally amuse myself by tossing out bogus sentences that sounded just like, say, Derrida, and people would nod in the same way.
That said, I applaud your open-mindedness. Postmodern criticism is full of the most unbelievable bullshit, but in retrospect I can't dismiss the whole thing. This sympathy would probably vaporize if I were forced to read those papers again. But I've learned about two things in the meantime that are reminiscent of postmodernism and which I do take seriously.
One is the recent wave of neural/psychological research about how our experience of ourselves and reality is constructed in ways that don't at all correspond to our beliefs. There are many such findings and from what I've gleaned, they consistently refute our concept of ourselves as rational actors making controlled decisions as we move through objective reality. (I'm still waiting for someone to write a good general survey of this material. There's a crying need for one, though it might not be very popular.)
The other is the history of religious and spiritual ideas, which I was astonished to find contain many of the same teachings about how our identities are constructed and conditioned. If you read any Vedantic literature, for example, the first thing you run across is "you are not who you think you are". Similar concepts are found in Western traditions. This skepticism about the artifacts of the conditioned mind is very postmodern in a way. But there is also a profound difference: while spiritual teachings are all about connection with something beyond mental constructs, the postmodernists deny any such beyond. Regardless of one's position on religion etc., I think this points to the real weakness of postmodernism: because it treats everything as the same kind of fiction to be deconstructed, it lacks any compass for distinguishing true from false or good from bad. It's inevitable that you end up with a profusion of arbitrary language, a.k.a. bullshit, when you start from the premise that everything is bullshit in the first place.
I know I'm late to the game here, but you might be interested in reading this:
http://speculativeheresy.wordpress.com/2008/11/26/the-semant...
It's a write-up of a talk given by author R. Scott Bakker, an almost-but-so-fed-up-with-it-that-he-gave-it-up PhD in Philosophy about the experiential consequences of neuroscience.
However, I will argue that this differs significantly from postmodern philosophy in one key respect: while anti-foundationalism certainly is a hallmark of the school of thought, it is never founded on objective scepticism, like science. Even the points made by neuroscience (and Scott Bakker) will have to be undermined by postmodernism, as any one viewpoint (or Meta Narrative) inflects any other viewpoint and has nothing behind it but the authority of people in power.
I have a BA in English, and to the best of my knowledge and training, postmodernism is a manifestly worthless endeavour. When concepts like knowledge, empiricism, veracity, and progress become nothing but whips in the hands of the powerful, you've effectively gotten yourself into a rut you can't get out of. As a result, English departments are dying all over the US. For a detailed look at this, get Jonathan Gottschall's brilliant "Literature, Science, and a New Humanities."