OP, I sympathize with you! I also perspire easily, especially when I'm lugging a few travel bags through an airport, dashing up and down stairs and running from gate to gate. I've often wondered, given that airports necessarily involve people running about and carrying bags, why airports aren't more air conditioned than they are.

I used to carry a USB-powered snake fan (primarily for use on board planes that lack air vents), but I found it inconvenient, only minimally effective, and somewhat embarrassing to use.

The best way I deal with airports is by traveling in light summer clothing, even during winter. I wear polyester or polyester/cotton shirts that don't show sweat (like you, people seeing me sweat only makes me sweat more) and that dry quickly after sweating. I always wear shirts outside of my pants to increase the ventilation of my upper body. And I wear oversized wrinkle-free cotton khakis for comfort both when moving and when seated.

I welcome any other ways of dealing with the heat in airports and other buildings. I spent the last week in Spain, and outside of the hotel room (where I could turn the air conditioning up high enough), I pretty much was sweating the entire time. Even inside the nicely equipped airport lounges, there was no relief from the heat anywhere.