Can Flonase cause your body to feel more anxious or like you have a little more adrenalin being released?

Unusual. Nasal steroid sprays such as Flonase are usually well tolerated. The most common side effect is nose bleeds. They don't cause a release of adrenalin. While anxiety is not a typical side effect, if the nose spray is stopped and the symptoms go away, that would suggest a link. It would be important to exclude other reasons for the anxiety symptoms.