Your story is quite similar to mine! I'm posting my experience here hoping it would help anyone looking for a solution.
So, around the year 2012, I was then using a Compaq laptop with a CFL (non-LED) screen. Oh, it was beautiful! I could look at it for a whole day and never had even the slightest trouble with my eyes. But then it sadly broke, and by that time they had stopped manufacturing laptops with thick CFL screens. What's more, I wasn't at all aware that there would be any significant viewing difference between CFL and LED screens.
So, I bought a new HP laptop, and a very cheap one to my utter regret. The very first day of using it, I had a mild headache and my eyes felt strained. I thought I was sick or maybe stressed or something, but just a few more days with the new screen and I was 100% sure it was causing my headaches and eye strain. I kept my old laptop and the new one side by side, and I could clearly tell the difference. I just couldn't believe anything like that could be possible. I searched online, but couldn't find anyone else with the same issues, or even a solution to the problem.
I was scouring the internet for hours everyday, posted on every forum out there, but nobody could help. Then some people suggested that it could be PWM which is causing me eye strain and headaches. They said I should turn up the screen brightness to full 100%, and it should remove the PWM flicker, but no, that didn't help. Another person asked to install a PWM software to turn it off, that too didn't help. I thought if screen was the culprit, I could just connect my old monitor which never gave me eye strains to the laptop. But within minutes of doing so my the monitor also became painful to look at. With that I was sure it wasn't the laptop screen, it was the laptop's internals where the problem was.
More than a month had passed by then, and I had to visit an ophthalmologist and had to start wearing glasses for the first time in my life! Also by then I started seeing similar reports of eye strain from a few people coming up online. Discussions started and it turned out that Intel HD graphics was the potential culprit. I tested the hypothesis by uninstalling the Intel HD driver, and voila(!), eye strain was completely gone for the first time! However, disabling it made my laptop sluggish, I couldn't even start Photoshop, let alone using it.
My laptop had an Nvidia card too, but it was connected to the Intel HD graphics, not to the screen directly, and so disabling Intel HD also disabled the Nvidia card. And, that's where the solution to my problems was. I needed a laptop with the screen connected to Nvidia card directly, not via Intel HD.
So, I put up with the nasty laptop for a few more months, saved some money, and bought a laptop which supported Nvidia G-Sync. Intel HD Graphics doesn't support G-Sync, so in order for it to work, manufactures are forced to connect the screen directly to the Nvidia card while Intel HD graphics remains disabled permanently!
I bought an Acer Predator laptop with Nvidia 1070, G-Sync support, and a PWM free matt screen. It also has Acer's proprietary Bluelight Shield which works way way way better than any of the dozens of blue light reduction techniques I have tried. I have been using the Acer laptop for years for full days and haven't once experienced any sort of eye strain or headache. Simply amazing.
So, anyone that has similar problems, look for a laptop that has the screen connected directly to the Nvidia or AMD graphics card. And to ensure it's built like that, it has to support G-Sync (for Nvidia) or FreeSync (AMD). If the laptop also has a PWM free display then that would be a cherry on the cake. Head over to notebookcheck and read the reviews, they test PWM for every laptop and smartphone.
Hope this helps someone!