"...maybe she feels awkward in social settings being a single woman sitting with an obviously married man."
This bothered me when I read it, but only now did the penny drop. If it feels awkward being an apparently single woman with an obviously married man, then you have forgotten or are ignoring that you are essentially a side piece, no matter how loving the relationship.
As a perennial side piece, if I found myself wanting a ring to stave off social disapproval, I'd be looking for a relationship that wasn't asymmetric, or at least one where I was the primary partner. Because being clear about your place and carefully not treading on wives' toes is pretty much the foundational part of being in a relationship with a married person.