First thing you should do, as Tapeworm suggested above, see if you have the same problem simply playing files from the internal memory with the card removed. If that suddenly fixes things, then try using a different card. And if the problem comes back, then yes, it’s may well be a hardware problem with the card slot and you can either just use the internal memory or think about a replacement.
But it’s worth a shot at replacing the firmware, which is what’s running your Clip Zip the way Windows or IOS is running your computer.
The Sandisk firmware is here.
http://forums.sandisk.com/t5/Sansa-Clip-Zip/Sansa-Clip-Zip-Firmware-01-01-21/td-p/297767
Follow the instructions for manual installation about halfway down–the Updater is unnecessary.
Download the current 1.x zipped file onto your desktop, un-zip it (Extract it in Windows), connect the Clip Zip and just drag the files onto the Sansa Clip Zip driveletter. When you disconnect the firmware will install itself in under a minute.
See if that fixes the problem.
If you’re going to use Rockbox, you really do need to follow the instructions there step by careful step. Basically what you are doing is installing a new operating system in the Clip Zip (though you are not deleting the old one) and it’s like changing from Windows to Linux–there’s a learning curve. If the Rockbox site is too computer-geeky for you, then that may not be a good solution for you.
Unless all your music files are m4a files from the iTunes Store, you could replace the ZIp with the Clip+ , which doesn’t play m4a files (that’s the big difference between the Clip+ and the Clip Zip). As I’ve mentioned elsewhere, I’ve gotten a refurb Clip+ from this eBay dealer that looks untouched and works fine for all of $20.
