Dogs, like humans, can develop allergies to anything at any point in their lives. It could be that since your dog is now a canine senior citizen, and her immune system is not what it was when she was in her prime, that there are things in her environment that are causing her to have mild allergies.
The snorting thing is VERY common to brachycephalic breeds (breeds with pushed in faces), and even dogs with normal facial structure have this from time to time. It's called a reverse sneeze, because it basically sounds like just that - a sneeze happening in reverse.
If it gets to the point where she is really bothered by it, contact your vet and ask if it would be OK to give her a bit of children's benadryl. If it IS an allergic reaction to something, the benadryl should help her out, but as with everything, check with your vet first before giving it to her.
Ghilly
The Shitzu has facial features similar to Pugs and English Bulldogs. Their nasal passages are short and constricted as compared to others breeds.
When they have congestion they do what you are decribing to clear their nose. Bullies do it all the time usually after waking from a nap.
I am surprised that your dog just started doing it now at 12 years old.