It's the salt in sweat that irritates eyes.
Water can get time to evaporate, but salt remains on the skin, and accumulates.
You don't feel it, but throw some bottle water on your dry forehead when back home, and a whole shitload dissolves in the water and floods your eyes.
What worked for me, though it may read far-stretched, is metal, as ehm heatsink, to decrease transpiration.
At times sun feels "hard", I put on a cap that I put some foil in the front part, and it made my forehead transpire way less.
A polyester t-shirt: I cut out a piece of foil, folded over the edge around the circumference alike they do for seams in clothes, and with needle and thread here and there 3 loops, just to hold it in place at the inside of the back of the shirt. It was end of sticky and heavy wet back of t-shirt.
My bike shoes: at work, from a pair worn written off work security shoes, before throwing away I molested the soles off, there are sole-shaped thin flexible steel sheets inside, to prevent sharp objects going through the sole. With a sharp knife I cut off the adhesive on it, bare blinking metal, and just laid these in my bike shoes. Bare skin on bare metal. And bingo, end of wet socks that show white salt drawings, end of sore feet. Despite hours feet even come out nearly dry. The shoes was were I got the idea for aboves 3.
Less transpiration = less salt = less irritation.
