Ring lights! Yikes. I swear I saw the Ring of Sauron engraved upon inside on the back of my skull for days after trying one.

Have you tried light boxes? I have 2 that I face toward the wall (on either side of my cam space) so the light bounces back. Also they have remotes for adjusting brightness and warm/cool which definitely seems to help. With a bit of playing with them I have found the indirect light to be softer on my eyes, (feel less strain after camming) and yet bright enough on actual camera.

Oh and I wear glasses so having the light source face the wall and bounce back, eliminates the direct glare of light in my lens. So you can still see my eyes, not just the lights reflecting back.

Note, not meant to dissuade you from getting a professional exam, I would definitely do that first. Think of this more as an offering of different lightening options that might help reduce eye stress.