iTunes set up the program so that you could only have your songs on one computer... the iDump is a good idea since you can't transfer to your comp. Kinda shady though. In the future you can download songs to your comp and should be able to share them with one other device, your phone. From there I recommend burning all music to a cd as that's the only way to get it from one comp to the next.

Sorry for your troubles. iTunes and Macs are crap.