Sorry for the delay, I did the experiments right away but didn't have time to report back.
Your argument that there is little chance for significant expansion sounds reasonable.
Ok, dry themometer I did the following:
Wide open mouth raised temp. significantly and quickly (so we know the response time of the themometer is quick enough to do the next experiment.
Puckered lips 1 to 6 inches away - No change in temperature.
(This too suggests that there is no adiabatic cooling of the air by expansion. I believe if the air were cooled, the thermometer would say so.)
Fan a dry themometer with a piece of stiff cardboard - No change in temp. This seems obvious but I did it for completeness.
Conclusion: Fan a dry thermo. or blow on it with puckered lips and get no change in temp. so temp of the air is not changing significantly.
Blowing on your hand or fanning your hand you feeling cooling but that must be due to evaporative cooling. I would have tried a wet bulb thermometer but the portable ones didn't have enough resolution to bother with and water was a long way away from the dry thermometer.