I'm doing the exact same thing.
Here is what I did.
First, The way things work is that if you have two tracks and there is a clip i both of them, the topmost clip will be visible. If you want the bottom track to be visible during a specific period, you must make the clip on the top track go away for that period. There are two ways to do this. Both involve using the razor to make two cuts in the top-most clip, one where the bottom clip should start to be visible, and one where it should end to be visible. After you have done this, you have "isolated" the part of the topmost track which you want to remove, to make the bottom-track visible. There are two ways to remove the clip:
1. Remove the clip.
2. Hide the clip, i.e. click on the little "eye" at the left of the clip in the track. A red cross will appear over the eye, and the clip will not be visible.
Your transitions will now work as you intended.