Instead of "charger," you should call it the "USB cable" to avoid confusion. When started reading your post, I was thinking, how can your son break a USB power adapter (the "charger" that plugs into a wall socket) by biting it...
Since you got the product with the two USB cables at the Apple Store, you have the right replacement (some third-party replacements are for the 2nd gen shuffle and do not work with the 4th gen shuffle). Make sure you are completely inserting the plug into the headphones jack. The fit may be tighter than expected, since the connectors are new.
If you tried both of those cables, it would be unlikely that you got two faulty cables in one package. If a new USB device has been added to the computer between the time it worked with the old cable and now, there could be a problem on the USB bus (such as not enough power or a device conflict).
As a test, you should shut down (power off) the computer and disconnect all USB devices (including any hub), except for standard keyboard and mouse (if used). Do this to reset the shuffle
Start up computer, run iTunes, and connect the shuffle by itself to a direct USB port on the computer.
Thanks for your reply. The cable is connected firmly, and I am using a laptop computer with no other devices in the USB bus. My computer doesn't recognize my shuffle, and I cannot get the shuffle to stay charged. And, iTunes doesn't show it either. I am going to try to charge the shuffle to my desktop computer tonight. Otherwise, if it cannot be figured out here, a trip to the apple store will be next?
Trying the other computer is a good idea. If that does not work either, then you either got two faulty USB docking cables in that package (seems unlikely) or the shuffle is not working. It's possible that the old USB cable that got damaged had a short, and trying to use it with the shuffle to see if it still worked caused damage to the shuffle.
This is definitely a 4th gen shuffle, correct? This document shows the four types.
If it's the older 2nd gen shuffle (which looks similar), it does not work with a USB cable that is for the newer shuffles (3rd and 4th gen), because the connector is different.