I am creating an Android app in Eclipse, but when I drag around a TextView, everything below it moves around as well.
This is when I haven't dragged anything.
https://i.sstatic.net/Twp4L.png
This is when I drag the TextView just a bit.
https://i.sstatic.net/oJ75k.png
The layout designer doesn't work very well in Eclipse. You could try Android Studio's layout designer (the new early access preview), that one is a bit better, but not by much.
The most common practice is to use the layout designer to only get started with and perhaps to use the Outline view panel to tweak some of the nesting of the layouts/views by dragging some of the nodes inside it, but then it's to dive directly into the xml code yourself. There is really no other substitute for doing that. The tool just isn't very good yet.