Hiya - sorry about the delay.
I have been tinkering a lot with this, and have come to the conclusion that in some cases it really does save time to start from scratch.
While I was unable to alter your existing code to prevent overlay (or entirely sort the z-index) it was (relatively) straight forward to roughly approximate your design.
I attach the working files created in this process. I'm sorry I didn't come up with something easier; I'm sure someone else could and hope they will!