Icon shows "charged" when not plugged in (2.)

Well,

I just had talked about this problem here:

Icon shows "charged" when not plugged in

But this was just a temporary solution. If I use the hardware soft removal from Windows after it has been charged, it shows the normal battery. But if its empty after using and I charge it the "charged" icon is there again.

And I have the impression that I can't charge the iPod completely full, it's just up to 80%. And I tried to use another battery, it didn't work.

I also resetted it a few times with iTunes and started it again. And I tried to replace the headphone jack hold switch cable, but it didn't fix it.

Another thing I maybe should mention is that the screen makes a weird sound when I charge it, it something like a gentle high endless beep, and it's getting higher when I turn the backlight more bright.

So what should I do? Is it necessary to replace the motherboard or the screen? Because this is too pricey for me...