That's me to a tee when I work out--my face is beet red and huge flop sweats. My hair gets so wet from sweat, it looks like I just stepped out of the shower. And I'm in fantastic shape--really the best shape in my entire life. I also have very low blood pressure, almost too low.
I think some people (usually people with pale skin) are just naturally prone to getting red in the face. Not only is my face beet red when I exercise, but I flush very easily. When I'm a little warm, when I'm embarrassed, when I'm excited, when I exert myself even the tiniest bit--my face is immediately bright red. People commenting on it will make my face get even redder.
There's a line in Pride and Prejudice about the main character's eyes having been brightened by exercise--that's the way I try to think of my face; it's "brightened" by exercise. I try to think of it as a "sporty" look.
I do see lots of folks around my gym with the same sweat stains that I have but I don't notice as many red faces. The sweat stains are easily dealt with by purchasing work-out wear that doesn't show sweat. There's nothing I can do to camouflage my red face (or my dripping wet hair), though.