Put a teaspoon of baking soda in the water when you cook them. Also, the less pointy part of the egg will have a small bubble in the bottom (the fresher the egg, the smaller the bubble). Tap the rounded edge first, gently, and you should have a small space that you can get a finger into. ETA: try to grab the shell and the inner membrane together.
It's likely that because you're cooking them in a basic (ph, acid vs base, etc) solution that the egg is being changed chemically. Using a harsher base might shed more light on that (though would possibly make it inedible).
