My first thought when I began reading this thread was: it's the battery! I would suggest that you charge the battery from the MAINS not the usb, using Sony's own charger or a PSP charger (the type with a round barrel). Leave it charging for about 12 hours.
However, if, after doing this, you find that the battery runs down quickly, seriously consider replacing the battery. This is what I had to do with my 505 - or at least a techie friend did this for me. There's lots of info about how to do this dotted around MR, and a good vid on youtube, made by the people who sell the batteries and tools (though I bought mine from Amazon).
Here's the url for the video: