I was unable to replicate the said bug on both mac offline version and online version. I dragged multiple text labels without editing them and then could select all individually and edit their text content, font, positioning etc with no problems.
We did notice a bug that if you remove all the text from the text field, then its impossible to reselect the text element.
I proposed a solution to this in the form of having a hierarchy of UI elements below the units section (ala photoshop) since that area is unused anyway. In this area you can quickly select each element, change their properties, delete or copy, and reorder them to different layers.
Here's a mockup, let me know what you think, whether this would be useful or not