Is there a psychiatric problem that could cause headaches and make me see things out of the corner of my eye?

These are separate. Headaches are a neurological problem by definition and can be caused by or worsened by stress; stress is not a psychiatric problem but a part of life. Consider seeing a primary care provider or possibly a neurologist for recurrent headaches. If you are concerned about what you see out of the corner of your eyes, discuss it with your doctor.
Anxiety may include. Excessive ongoing worry/tension, an unrealistic view of problems, restlessness, feeling edgy/irritable, muscle tension, headaches, difficulty concentrating, sweating, nausea, frequent need to go to the bathroom, tiredness, trouble falling/staying asleep, trembling, & being easily startled (with sense of seeing things out of the corner of the eye that disappear upon looking). Seek evaluation.
Multiple causes. I absolutely agree with my colleagues about the need to rule out possible neurological causes of these symptoms. That said, extreme anxiety and stress could lead to tension headache, and certain types of anxiety can lead to hallucinations. Another thing that comes to mind is extreme sleep deprivation. Do see your primary care doctor first. Good luck to you.
Not psychiatric. But neurologic. They may be different entities all together, but they coul be part of partial complex seizures. These seizures do not cause falling, but do cause perceptual disturbances. There are usually more symptoms though. You should see a neurologist.