The touchscreen is now working. I'm using the internal pull-up resistor on the ADC pin to detect when the touchscreen isn't touched. The algorithm looks something like this:
uint16_t x = touch_get_x();
// ADC values go from 0-1023, so a value over 1000 indicates
// that the screen isn't being touched and the resistor is
// what's pulling the value high. The value when someone
// is touching the screen varies from around 100 to 900.
if(x > 1000) {
// Screen is not being touched
} else {
// Screen is being touched
x = touch_normalize_x(x);
// ...
}