I started out similarly over a year ago, OP. I was as red as a tomato! Now I can go longer, further, harder, and finish with only a slight flush.
I still sweat like a mofo, but now I wear it as a badge of honor. I've knocked off 50+ pounds with that sweat, so it's nothing to be ashamed of. I keep a towel or two with me at the gym, along with water and good music, and that helps.
If you get upset at other people at the gym, use that anger to push you! Anger is an awesome motivator for the gym

Take things from your personal life, and use that to get pissed off at the gym. Good loud music helps too
However, maybe you are pushing yourself too hard, too soon? If you can't sustain the momentum for 20-30 minutes, then you need to lower whatever you're doing down a notch or more.