This afternoon at work, I returned to the department after being outside with a kid. I was tracked down and told one of those things you never want to hear, "Your husband called. It's an emergency." I got in touch with E, who said, "It's Cosette. I think she's had a stroke."
Even though I was shaking and on the verge of tears, I managed to call the vet, whose office closes early on Thursdays. They said if E could get her there soon, they'd stay open for him. I informed them that he is severely allergic to cats, which poses a problem as they have a resident feline. (Hence the reason why I do all of the vet excursions).
The short ending to the story is that Cosette most likely did not have a stroke. She seems to have a vestibular issue going on, possibly a side effect of the antibiotics she's been taking, or quite simply what is called "Old Dog Vestibular Disease." That would explain the falling over, walking in a circle, and head tilt E observed. So, we're trying some new medication and watching her. E occasionally suffers from vertigo, so he relates to what she is hopefully going through (versus something more serious).
Another clue that Cosette probably did not have a stroke it that she hasn't shown any personality changes. She's sleeping now and we're keeping a close eye on her. I don't know how you parents, with actual human children, do it. I get pretty worked up over my four-legged ones.