Am I going about learning to draw the wrong way?
You say


  Things in the real world aren't 2-d, they're 3-d objects. I'm not
  making a 3-d object when I draw, I'm making a bunch of lines on a 2-d
  piece of paper that will create the optical illusion of being a 3-d
  object.


and you are absolutely right. That's what drawing is.

If you want to start thinking of 3D objects as 2D I suggest you  use only one eye when you draw or generally examine objects.

You can also take pictures of objects and draw from them (with the help of a grid on top, if needed). Or, better, place a grid frame and draw looking through it (this technique was previously used for acquiring accurate perspective, but I think it will work well to get you to think of 3D objects as flat):


*try it for smaller drawings though (A4 or A5) and without grid on the actual paper

Getting familiar with the basics of perspective would also help a lot.

Where shade falls shouldn’t be too confusing to you if you generally think of objects as 3D. If you don’t get too technical about it, light beams travel straight from the source of light and illuminate the surface they fall right on. The rest is in shade.

About sculpture, it is not necessarily an expensive undertaking. You can dig and refine your own clay. Plaster is relatively cheep too.