I also sweat a lot - and my face turns beet red
.
Really, just try not to care
. It really IS just a sign that you are working hard and you can't control how much your body sweats.
If the sweat patches on the shirt are really bothering you, try wearing something with a lot of pattern - it kind of hides the wet bits. A friend of mine hates the sweat on her back so she tried putting deodorant (antiperspirant) there. It seemed to help a bit, but I'm guessing your body will find a way to get rid of the water elsewhere
.
Also, I don't know much about this, but aren't there shirt (like Under Armor) you wear under your work-out clothes that soak up the sweat? That might be worth an investigation.