If you touch cotton wool, do you feel hot or cold? I don't think you feel anything, until and unless it's wet.
So, you see that cotton wool is a bad conductor of heat, and it doesn't give you any feeling about its temperature until and unless it's wet, but even in that case, you'll be feeling the temperature if the water and not the cotton wool.
Sensation of temperature or heat, that is, thermal stimuli in Biological terms, truly comes from whether heat flows from your skin to the body or not. If heat flows from your body to the other body, the temperature of your skin gets lowered and you feel cold, the nerves get stimulated by bulboid corpuscles (end-bulbs of Krause) and send signals about the change in temperature, and your brain interprets that the body is cold. Similarly, when you touch a hot body, heat flows from the body to your hand, which raises the temperature of your hand, and your brain interprets that the body is hot.
Just like cotton wool, dry wood will also not give you any such feeling of hot or cold, until and unless it's humid.