Most actors are quite shy, but as soon as they get on stage they 'become' the character, so the shyness disappears. If that isn't happening for you, I think you just have to do more of it, so it becomes second nature to you and the audience just won't matter.
As for the private lessons, no, they wouldn't help. I'm sure you'd do fine in a one-to-one situation, as that isn't your problem. Working with a medium sized group in an acting class is a better idea. That's why most successful actors have done a 3-year full-time course at drama school. After three years of classes, several each day, you'd have all the confidence necessary to really give your best on stage. If you can't do that, at least join a long-term class, at least weekly, and attend for many months or a year.