Yes, you will get a new cable. Yu can do it at an Apple store
Make an appointment at the Genius Bar of an Apple store.
Or send it to Apple
For the above you enter the serial number fo the iPod since the cable does not have a serial number.
I have had problems with my iPod touch 5 g too I have had it ever seens it came out and it always work fine intill the new update came out 1st it was wifi wouldn't work then I updated it to iOS 6.3 or what ever it was then iOS 7 came out and I didn't updAte it right away and then my ipod wouldn't charge at all and I tryed like 6 other chargers and it wouldn't change then I updated just to see it that would do anything and it did it charges fine now and I'm no the only 1 that these thing have been happening to samething happen to my brother and sister they both got a ipod when I did and 3 of my friends that got the ipod around the same time i did the samething happened to all of them I think Apple is making us update or the 6 of us got messed up iPods just try updating it if your not on iOS 7