Yes but from the sun, not from the heat in the air molecules. And if you were not shielded from the near vacuum, you’d cool down from the evaporation of sweat on your skin.
At the equator, on the surface of Earth, solar radiation is about 1kW per square meter. According to this¹ wikipedia article 55-60% of solar energy is lost on its way through the atmosphere.
So would ~2kW / square meter at thermosphere height be enough to make you feel warm-hot despite the heat lost to evaporative cooling from sweating?
One might sweat a couple of liters/hr, and thus might manage 2 kW. But that's not needed. Between bright hot Sun, too hot, and dark cold deep-space sky, too cold, Earth spins, mixing too hot, and too cold, into not too bad. And so can you - the "barbeque roll" thermal management strategy for satellites. And if you're head-on to the Sun, or in shadow (holding an umbrella)... I don't recall whether radiative loss is sufficient to shed basal metabolic rate without sweating or not. If not, you might build yourself some elephant ears, and sleep with them edge-on to the Sun (and Earth). But EVA suits use water sublimation cooling - because people do EVA to strenuously exercise, even in the bright sun.
Science education content, down to kindergarten, mentions Sun heating Earth. But pervasively fails to mention deep-space sky cooling Earth. So a lot of explanatory leverage is left on the table - "Why are nights cold? Especially with clear sky? Especially in the desert? Why are mountains snow-capped? Why is winter colder?" etc. I wish I knew of a forum/community in which to discuss and create such improved content, but I've been failing to find one. :(
The air in this case is what's at high temperature so it wouldn't be drawing anything away at all. You'd still lose heat through infrared radiation. I'm not sure which effect would be greater.