Hello, I wanted to summarize what helped me so far and what might solve the problem, or some parts of it for good.
I have been diagnosed by an optometrist (not mistaken for an ophtalmologist/eye doctor) to have eye convergence excess and resulting accomodation insufficiency. That means my eyes do not work together properly to hold a single image instead of seeing double and I am already putting a lot of effort into that in real life.
Together with this my visual system (not eyes, but brain) is very sensitive. This might be the cause of the problem I have with modern displays - in some yet uncomfirmed way they might disrupt vision, causing my system to work very hard to keep a clear and singular image/text. That leads to eyestrain, overload, headache and migraine.
The help for me first were glasses with prismatic lenses, that ease of the problems by about 50%. I can see with them clearly only up to one meter and I was recently using an iphone 6plus that was before unusable for 2 or more hours with a slight discomfort and very small to zero headache after.
Another help is doing a visual therapy, that should make me able to use my binocular vision like a healthy person. That consists of about 6 months of 30mins daily exercises using - brock string, switching various lenses while reading to learn to adjust accomodaton, and tranaglyphs with 3d glasses, that I have to keep connected in one image while they move further and futher apart.
I hope that this solves the problem. Regarding flicker sensitivity I do not know if that helps, as noone knows which individual devices really do flicker and which not and how much. Its up to everyone to test with their own measuring tools. There is a possibility that flicker is disrupting the vision and when you get it fixed, it doesnt bother you anymore.
You can probably do a little test of your own with the brock string - either find one or make one. Then when focusing on the middle bead (2nd), try to be aware of seeing double the closer(1st) and the farther(3rd) bead and the string too. Then, the string should meet in the bead youre focusing onto. If it crosses sooner, you have convergence excess, if it crosses behind it, you have the opposite.