If you shave balls it itches because the hair just started growing again and since it is so short its not that flexible and this causes it to simply itch really bad.
There are different things you can do against this. If it itches right away after you shaved the area there is something wrong so you should probably wait with the next shave and use some conditioner or moisturizer for the skin.
If it itches after one to two days you should really just consider shaving again. I recommend shaving every day anyways since this gives you the practice that youll need to do it quickly and easy in about 5 minutes (after your morning shower for instance).
if you still dont want to shave again you should use some conditioner which will cause the hair to soften and therefore not to itch anymore.
Another idea is to use hair remover cream which also helps with the itching or the already growing hair.
It is also always a good idea to use moisturizer which is even good for the skin.
So if you shave balls really just do it daily or each other day it makes it way easier for yourself.