Greetings. I just got a new iPod Classic 120 GB to replace my 6-year-old 2G, which finally died on me near the end of 2008. I, too, have experienced the irritating skipping of songs. I'd also gotten a new Mac Mini and updated to 10.5.6 and update ALL of of the software, including iTunes, which is 8.0.2, I think.
The backlight always on "fix" didn't work, either. Then I restored the iPod. First I clicked on "Check for Update" (it told me that I had the latest available), and then I clicked on "Restore." After putting 3300 songs back on the iPod, I went back to one of the songs that the iPod would skip after just :03, and it played fine. However, just as I was beginning this paragraph, after 20 minutes or so of skip-free play, another song cut out prior to its scheduled end.