I would suggest adding your audio by dragging and dropping on the exact frame where you want it to start, rather than adding to the background. (Well, it does not have to be "exact" because you can adjust it later. You just want it on the clip instead of in the background.)
The audio should appear below the video strip, instead of in the background.
I tried out what you said your problem was and realized that I had the same issue when I tried to have two sound tracks on the same project. I messed around with it for a bit trying different things a discovered that if you have the second track on the project like this
instead of like this
You can use the slider to adjust the volume. To do this you just drag and drop the sound track over the video clips instead of the background.
Hope this helped.