Your screen-related headaches sound kind of similar to mine, which after using a 'bad setup' almost always starts in my left eye. If I push quite hard into the nose at of the top of my eye then the pain is relieved a little but comes back when I let go. The pain spreads from the left eye into the left side of my head and is accompanied by nausea and can last for several days also. The pain itself is relieved with sumatriptan (migraine medication) but this also leaves me feeling pretty groggy.
I can also get headaches at the top of my head which last a day and can be relieved with fairly high doses of normal painkillers.
The reason why I wanted to mention this was that from reading your post you may have the same problem as me, which is a sensitivity to temporal dithering, a type of flickering that makes devices appear to display more colours than they can actually create.
Why this is relevant to you is that if you can watch movies on your TV, it is unlikely that it is the TV itself that is causing you the eyestrain, but the source of the games that you are playing. For example, I have a TV that I can watch movies and programmes on all day long. However, plug in an Xbox One, or a 'bad' laptop via HDMI that uses temporal dithering, and I get this same migraine / headache that can last for days.
So I would seriously look into the system that you're playing games on as these problems tend to get worse over time if you keep persisting with a 'bad setup'.