Hey I have the same problem with my 4th gen nano, I pulled it apart yesterday to replace the screen and it looks like the cable has been corroded on the top 2 pins. We also considered that the logic board was failing to provide power to the screen, but seeing as you have the exact same symptoms I doubt its the power supply. Unless you can replace that cable I don't think the ipod can be saved.
I defiantly think that the cable is the problem here, does anyone more knowledgeable know if the cable can be replaced? if so where could I find the part?