How "thick" should the batter be? When I make this it crumbles and falls apart...

Mine's usually as thick as heavy cream, especially after it rests, which is important.

You might have a couple of things going on. Get yourself a handy-dandy oven thermometer. Your oven might be too hot. Otherwise you just need more fat. If you find that your batter is too thick and need to add more water, add a little bit more olive oil too. The water evaporates quickly, the olive oil will hold more moisture in your finished product.