In Fallout 3 & New Vegas, you could pick up an item by the part of the item your cursor was pointed at. IE, the tip of a gnome's hat so you could place the gnome upright.

I thought this was how the Skyrim used to function, but after playing the game again with Dawnguard, it seems I'm not able to do this, and thus trying to decorate my house with a statue of Dibella is essentially impossible. Has something changed, or am I taking crazy pills?

The items are separated into parts now. If you pick up a bowl, it will turn depending on it's weight.

If you pick up a leg, it will depend on whether you clicked the calf or thigh, and so fourth.

No, you're not crazy.