Multiple shift registers, 16 bit numbers
Your problem is entirely that you declare val to be uint8_t, instead of uint16_t. You are truncating your data to 8 bits before displaying it. One other note, the 74HC595 has a maximum total drive current of 50mA. Your circuit will draw around 90mA for each of the 74HC595's if all of the LEDs are on. You may want to change the resistors to something bigger (510 Ohm, for example) to avoid this problem. Of course that will make the LEDs dimmer, so it's a tradeoff. Here's your project in 123D Circuits. You can simulate it in the browser and look at the code running in the Arduino there.