G4 Randomly Shuts Off

I have a major problem: Over the past 8 weeks my Power PC G4 tower randomly shuts down. I cannot copy a CD, transfer files to another external hard drive, and certainly cannot burn a DVD or CD without it turning itself off. It seems to shut down faster when it is doing something related to A-V files, but if I just leave it alone it may shuts down. Shut downs happen from a few seconds to 30 minutes after I restart.

I have changed the PMU battery and reset the PMU chip to no avail. I am very worried, because I am planning to buy a new iMac when they are updated, but I don't think I will be able to migrate my data. It seems like a few years ago I had this problem (but not as bad), and was told to delete certain preference files, but I can't remember. Any help would be appreciated. Thanks.