You sound like a terrific friend, and like you really value so much about your friendship with this person. Maybe the whole thing - dog getting old, denial about losing your dog, guilt, fear- just makes the whole thing awkward all around. People get weird around death and grief it seems.
If I were in your shoes, I'd contact her one more time to find out what's up (on the sitting, on what to do). No not your responsibility as you've gone beyond the call of duty for a friend but what the heck, it's just a call and it sounds like you value your friendship and the dog sitting (or so that would be my logic if I were you but only you can weigh the cost and value of calling or not). You obviously don't want that responsibility (and I wouldn't blame you!), and she maybe (at some level) is dreaming of also not having that responsibility or avoiding being there when the dreaded day comes, and she might feel hurt/judged because you were so straightforward (understandably).
On a more practical note, maybe the compromise should be you could take her to the ER or vet, you have a way for the vet to reach her, and the vet talks directly with your friend to make any serious decision about the dog's health. Almost no where in the world these days someone can't be reached by phone.