We know that Fairy does normal damage to Grass. If Exeggutor is Grass/Psychic and Fairy Wind does normal damage to it, we can conclude that Fairy does normal damage to Psychic.Uh I don't think the Exeggutor being grass/fairy has been disproven. Doesn't seem likely but it's not impossible.
We know that Psychic does normal damage to Fairy. We know that Psybeam is NVE on Exeggutor. We know that Psychic is NVE on Psychic.
By deductive reasoning, we conclude based on the above that Exeggutor has not been retyped to Grass/Fairy, Exeggutor is still Grass/Psychic and Psychic does not resist Fairy.