Sensors for detecting breathing movements

Temperature is a measure of the average thermal energy. A temp sensor will not tell you if we are talking a dry or wet bulb temp. The temperature is the same for both.

Wet or dry comes into play when you start thinking about how much thermal energy would be required to change the temperature of the air or how much condensation you will have if you lower the temperature of the air, etc.

To measure wet bulb, you need temperature and relative humidity. With those two sensors, you should have what you need.

From your explanation and other question, it seems you are connecting your IR LED directly to the ADC input. I don't think this will work too well at a distance, the ADC input will probably have quite a low impedance that will attenuate your signal. Photodiodes have a very large impedance so you need a transimpedance amplifier to convert the current to voltage.
I would use something that is designed for sensing rather than emitting, like a photodiode or your IR phototransistor (if it didin't work then you are probably not using it correctly), and feed this into an opamp, then into the ADC.
In the app note you link to, there are plenty of example circuits, all of which involve a transistor or opamp to amplify/buffer the signal. Try one of these and see how it performs.