I do my commuting in the Southern humidity which is tantamount to riding through a steam room like your uncle Morty used to use to do business meetings.

The whole time I've been here, I haven't yet found anywhere that thinks twice if I excuse myself to the men's room for a minute or two so I can clean up. I'm polite about it, I don't just ball out and run to the restrooms. I'll excuse myself like "Hi, good to see you... hey, I'm gonna clean up for a second, I'll be right back." When the humidity is seriously nasty, like it's about to be this month, I will carry an extra tshirt in my backpack to put under whatever it is I'm wearing. I also throw some wet wipes and a washcloth in there. It makes it so you can wash up and get back to presentable in a flash.

I'd be hard pressed to imagine there's any workplace that would mind if you got to a meeting 5-10 min early and cleaned up real quickly.