I wonder if your doctor believes that you are having great tension (feels like grinding to you) headaches. These are pretty common and sometimes a level of mental health that isn't coping well will lead to more tension headaches. Maybe this is why your doctor put you on antidepressants? (what medication are you on?) Walking is a natural reliever of stress. It lifts that pressure but you still feel the tension Musculature in the head is extensive. One area is tight, you feel it in various spots. But over the eyes is often tension.
I will tell you that this kind of pressure and sensation you describe IS secondary often to depression. (meaning depression is the root). I can only say that in general terms as I don't know you and am not treating you but your doctor is. Depression can look classic or non classic. But again, I'm not in the 'know' for your symptoms or why the doctor diagnosed you as such. Worth a conversation with them as to WHY. And if you still contradict that, further discussion. https://www.medicalnewstoday.com/articles/depression-headaches
Perhaps your doctor actually prescribed antidepressants because you have anxiety. They work for that too. https://www.healthline.com/health/head-feels-heavy
I don't mean to focus on mental health. I know that isn't what you were looking for. But wanted to explore it further with you.