Yep... I have the same problem although it sounds like you might have worse physical effects. I mostly avoid speaking in front of groups these days, but I have gotten better over the years.
I recommend talking to people about your fear and letting some people in your class know that you have this fear. I found it a tremendous relief to just get it off the list of taboo subjects... I CAN'T SPEAK IN FRONT OF GROUPS. So what? Do you know how many people share your fear? Almost everyone on the planet, but you are one of the unfortunates that gets physical effects.
So... the idea of being open about your problem makes it easier to deal with when you start to have effects. If I started to shake I would just stop for a moment and take some deep breaths and possibly put my hands in my pockets or on something solid. If my voice started to shake I would take a drink of water... but be careful because if you're shaking the water drinking can be quite risky

As far as sweating and redness... just make sure you wear clothing that you feel will not show the sweat or redness as much. Try to use the clothing to minimize the effect. I have a friend who used to (and probably still does) get hives on her neck when she spoke in front of a group... she is a lawyer now and has to speak before judges all the time. What she did was wear a high collared shirt on any day that we had to speak.
I tried a lot of approaches and never really found a way to make speaking in front of groups a good experience, but I've come to not be embarrassed about it and that has helped. I now know that there are very few people who do not feel like they are in a fish bowl when they are speaking to a group...
Try breathing exercises and really know your material... BUT do not run yourself ragged trying to do the speech perfect. Sometimes I would run a speech through my head to the point where it was not letting me sleep properly and stuff. It isn't going to be perfect so don't put pressure on yourself to make it perfect... just get it over with was always my approach.