Hi Sven-Olof and Harold,
Thank you for good advice, I am confident now that my planned design will work.
Both of your designs securely fastens the horizontal dividers into the vertical dividers using a dovetailed dado, or through tenons.
I will plan on doing something similar so that the whole structure becomes more stable. After some thought I think I also will use stopped dadoes, leaving an inch of the vertical dividers with full thickness.

Thanks,
Birger