Your best bet is to try, and replace the screen. Here's a link to the guide for your iPod. iPod Touch 2nd Generation Display Replacement , and there is a link on that page for the screen. You can buy it directly from ifixit or find it somewhere else if you prefer, if your a little tight on money you might try eBay, but be really carful, check where it's coming from and read the entire description. It also looks like you can replace the power button. iPod Touch 2nd Generation Power/Volume Controls Replacement , but they don't appear to sell that part, you can try some other places, http://www.directfix.com might have it. I don't see anything for the home button, so you might be out of luck, i will see what i can find though. Links to all the tools you will need for the repair will be on the first link. Hope this helps!
try checking those songs in iTunes mine was doing the same thing and i went into iTunes and tried to play them, and it played the exact same amount that it played on my iPod and skipped to the next song. I just reimported the songs that it was skipping and it fixed everything. Hope this helps!