I know exactly what you mean, Baily. This happened to me as well when I first started. I worked so
hard over the week on what I was assigned. I could play it in the practice room. Then came the time to play in front of my teacher. My hands would shake, I would forget to breathe and I would choke up. My teacher would say something like, "First time doesn't count. Take a deep breath and play again....but this time... BREATHE!" There are still times I get nervous, but my comfort level has grown over the past 2 years and 10 months. I can play for my teacher as well as I can for the chipmunks that listen to me playing on the deck in the summer.
The problem is... you are an adult and used to being good at what you do. Now you are a "beginner" and you want your teacher to think well of you and your efforts to play.
That puts a lot of pressure on you. So much, that you get the rush of adrenaline that makes your hands shake. Your teacher has seen this before. It has undoubtedly happened to him/her. So try to take a deep breath before playing the first note.
Also... practice "performing". I used to sit in the practice room and imagine my teacher sitting there listening to me. Just the thought of that made my hands shake in the beginning and mess things up. But after a while, I got better at it.
Hang in there. Don't be too hard on yourself.
BTW, I have no problem speaking in front of a crowd of people. Put a guitar in my hands and it is a totally different thing. But I have been speaking from birth and feel confident that I can express myself. That is not true of playing music. So the only cure is to play as much as possible until you "recognize" the feeling of nervousness and can deal with it. You will find a way to cope. Being "prepared" gives one confidence. Having faith in your preparation and remembering that you are there to "share" the music will help.
When the sun shines, bask.
Classical Guitar forever!