Jess keeps on sneezing with running nose because she's been having nose allergy for years.
1. Shall I say running nose or runny nose?
2. Shall I say nose allergy or only allergy for Jess always sneezes as soon as she wakes up in the morning. I am not trying to dig into medical terms but just want to have it expressed in a proper way in English.
Merriam-Webster's Learner's Dictionary
I believe you can also say "My nose is running".
Verona_82, you're right.