With two years of skiing experience, you've already got a solid foundation, and that's important. Most people take about two to four weeks of intensive training to do proper carving turns and have them look halfway smooth - but honestly, that's more like your first taste of success. To really carve cleanly, I mean with proper edge control and without any skidding or drifting, it can take a couple of months, depending on how often you're out there. The key is that your body has to understand the feel of weight distribution and edge angle - that just comes with repetition.
With your all-mountain skis, you're not really doing anything wrong. You don't necessarily need special carving skis as a beginner - those are more for people who are already really confident and want to push their limits on the slopes. What matters more is that your skis aren't too wide (under 80-85mm underfoot is ideal for learning) and that they have decent edge sharpness. If you notice your skis getting dull or not holding, getting them sharpened at the ski shop can work wonders - that often makes more of a difference than new skis.
My tip: spend a day or two with a good ski instructor who'll actually show you how to properly stand on your edge and shift your weight. That'll save you real frustration and break some old habits. After two or three days of that kind of input, you'll notice yourself when it clicks.