I used to debate in high school, and I remember being terrified my first few times. The very first time I spoke, I couldn’t stop at all was my right knee shaking, I remember feeling like my leg was spasming. But afterward when I mentioned it to my friends, nobody noticed it. And it wasn’t out of politeness, my friends never miss an opportunity to make fun of each other :) So chances are if you feel like you are falling apart, it probably isn’t that bad. That being said, I agree with @martyjacobs, don’t hold light paper. Heavier stock paper, or paper pasted onto construction paper is okay, but you really can’t hide shaking paper because it’s loud.
To stay in one piece I would think of it as a conversation. I’m assuming that you are quite capable of talking to friends/family about things. Even explaining new concepts, persuading an issue, and all the sorts of ‘speeches’ you give in class. It’s seems corny to say, but imagine that the entire audience is just in an everyday conversation with you and you mentioned your topic X. They all said in unison “Hmm, I’ve never really thought about X, but that’s interesting. Tell me what you mean.” And then explain away, to your willing audience…they want to hear what you want to tell them.
As for remembering where to go next, make sure to really focus on your transitions in your speech. If your transitions are weak, it will feel more like you are jumping randomly from point to point, and it’s harder to remember which one was next. If you work hard at setting up a new point and leading directly into it, it becomes much easier to follow the flow of your argument. When it comes time to give the speech, I would immediately beforehand (2–3 minutes prior to) review in your mind the transitions between each topic so you have at least those down. Forgetting a quote, or skipping one detail is minor league, but stopping dead in your tracks as you try to recall the next point is more noticable. Good luck, and hope this helps!!