no,, the crop factor is the same whether a DX or FX lens. The DX lens means that it has a smaller circle and only covers a DX sensor, so if you use a DX lens on a FX body, then you would have a circle projected onto the FX sensor which is why when a FX camera detects a DX lens, it will automatically crop down the sensor to the DX size.
But both the FX and DX lens on a DX camera will fully cover the area of the sensor and both use a multipication factor of approx 1.4-1.5 depending on the camera. So, for example if you buy a sigma 10-20, that is the same as a 15-30 range.
Hence if you have the 55-300 DX VRII lens, that has a reach of a 450 mm on the D5100/7000 or any DX camera.
That is one reason there is a hard core DX following that don't want to give up that extra reach (think bird photographers).
The 18-200 is a really nice zoom, because it gave you effectively a 27-300 mm zoom in one package. Somewhat wide to fairly long tele.