Deep breath. Dogs don't die over runny noses, at least not instantly. This is doubly true if it isn't impacting breathing and is just clear and drippy.
She's young and many, many dogs are vaccinated late because they are strays or at a shelter or people forget.
Combining food won't cause allergies, but more ingredients mean more potential triggers.
Just to calm you, my dog gets a runny nose twice a year, in the spring and fall. She's 9 and she's been through one owner before me, 2 shelters before me, and 3 vets from me moving around. Every one has noted it and said so long as she isn't developing a cough, it's just a nuisance allergy and that most medicines would probably have more side affects than is worth bothering with. And that's even counting the vet who was basically a pharmaceutical rep. So for a few weeks I keep a box of tissues around and wipe her nose and wash the couch cover and her bed when she wipes it on those. I've been told that kind of runny nose, clear and watery, is fairly common for a dog with seasonal allergies.