Will cat allergies necessarily cause itchy nose/watery eyes?

Had such trouble phrasing this clearly.
I have a rash that I am concerned might be caused by a cat allergy. However, my nose doesn’t run or itch when I’m around cats, and my eyes don’t water. So I’m clinging to this for hope – would I necessarily have these symptoms if I were allergic? Or can a cat allergy be expressed as only a rash alone?
I know I’ll need to see an allergist to know anything for certain but I just want some input right away. I love my cats :(