Why, after fully charged, does it turn off while in use?

Lol on your son bugging you! I know the feeling x 5! Here, check a few things for me. First and foremost, look in your charge port. If this problem happens without a cable plugged in, then we can rule out cable. Now, look for the obvious (big clump of dust, grass, paper, etc.). What I do when I get one in is take my trusty hobby knife, with a pointed blade inserted, and GENTLY slide it, first, across the bottom of the port and then use a dental pick to pull out the, 99% of the time, clump that comes with it. Then I repeat the process, twice as careful, across the top. After I'm pretty satisfied that nothing more is coming out, I take a medium bristle toothbrush....unused of course....dip it in 99% isopropyl alcohol (rubbing alcohol, and yes 99% please) and gently massage the port back and forth, then put the bristles in and pull out. Hopefully this is all you need to do. Your problem is indicative of a plugged up dock connector. Hope this helps.

Is the liquid damage indicator activated? If you shine a flashlight in the headphone jack you should see a white/silver dot. If that dot is pink or red then it has been exposed to liquid and could explain the issue you are having. You say when charging it is easily disrupted. Are any of the pins in the USB port bent? That is a very common issue to have, especially with younger users. They often try and jam the USB cable in the wrong way and it bends the pins.

I would try doing a hard reset to see if that fixes the issue. Back up your iPod first