I can also confirm this multiple header break. I need to add a second header bar to replace the primary left header bar on tablet and mobile screen sizes, but as soon as I add another header bar it breaks all the formatting.
Below are some screenshots of what I’m experiencing on one of my sites running Update 1.2.1 -
Only one header bar
This is what the site looks like with only one active header bar on the left:

Displays correctly in Header Builder:
When adding another header bar within the Header Builder everything displays correctly:

Broken front-end display:
But when viewing the front-end, all the styles and formatting is completely broken:

As soon as I remove the second top bar, the formatting and styling returns to normal again.
Hoping we can get a fix for this soon as this site in currently not working on smaller screen sizes at all.