I think that it's normal for your diaper to be wet... As soon as your baby pees in it, it will get wet throughout, that is why you need to use a cover with the diaper. There is not enough fabric to keep the outer layer of the diaper from getting wet. You do need to change diapers often, every 2-3 hours is about the norm from what I hear, but with young babies you may need to stick closer to 1 1/2 to 2 hours. If you change your baby that often and every time the diaper is completely soaked (back and front), then you probably would need a liner, but at 2 months old I would be surprised if you needed one unless you have a really heavy wetter on your hands! But if you change him/her often enough and some parts of the diaper are still dry but you are concerned about the diaper feeling wet on the outside, it is completely normal and that's what the cover is there for!
Hope that helps!