How do I stop content overlap with Display Suite on mobile/narrow screens?
Found the solution when I discovered a difference between this site and the other that did not register fully when I posted the question.

The Content Types on this site and the Paragraphs on the other both use Display Suite, but this site was using one of the default Display Suite layouts. I built the other one later and used the more flexible Adaptivetheme bootstrap-style layouts.

Turns out the layout is the reason that items were stacking vertically on one site and stubbornly staying side-by-side on the other. When I switched this site to an Adaptivetheme layout like the other, things were much improved:



In case anyone else has a similar issue and is stuck with a default Display Suite layout, I also found that the email field was the only field causing problems. You apparently cannot truncate an Email field, but I found that switching the content regions so the taxonomy tags were on the left and the location, contact, and links were on the right was better than the original layout because the email address could stretch its box without running into the box on the other side.