Which form of the word scare would complete the phrase feeling of .. It might be something like afraidness. I don't think scariness works here.
There is the noun form scaredness, however it is not a common word like fear and "feeling of scaredness" isn't a useful phrase. Scare itself is used as a noun also, meaning sudden fear.
scaredness: the quality or state of being scared
scare: a sudden feeling of fear
"Feeling of scare" is somewhat acceptable but it is not a common phrase also. I see the usage in psychological contexts.
Example from the book "Integration in Counselling & Psychotherapy: Developing a Personal Approach" By Phil Lapworth, Charlotte Sills:
For example, the therapist may simply say,'You look scared right now' in response to facial and bodily clues of fear, in order to facilitate the client in acknowledging, contacting and expressing their feeling of scare.