In the Adobe Reader, there is an icon that looks like a square with an arrow pointing out of it. This is the "share" menu. Tap this, and choose "Open In". If you have dropbox installed, you should see that as one of the choices. If you don't see it, you may need to scroll the menu item. Tap DropBox and the dropbox app will be launched where you can choose the folder to which you want to upload the file.
Note that you won't see the annotations in Dropbox when you view them in the dropbox ipad app because the dropbox ipad app does not display annotations. However, you should see them if you open them on your Desktop or laptop using the Adobe Reader.
Please let me know if this helps!