Yes - it does sound like a hardware issue now so I'd go to an Apple Store. It doesn't matter that you haven't purchased Apple Care as it's under warranty and they'll swap it right away for a new one. But it might be worth calling in advance to check they have one in stock.
I'm sorry to hear that you've had this trouble - please don't dispair - it's not a typical situation and we hear very few reports of faulty iPads. Let us know how you get on at the Apple Store and if you need any further help don't hesitate to let us know.