This doesn't answer your question directly, but I make eggs fairly regularly and I have a T-Fal non stick pan dedicated to omelettes and eggs fairly exclusively. It never gets put on high heat, never gets rinsed hot and is hand washed (which takes about 30 seconds really).
I struggled for years with other pans to do eggs, and I found out lots of chefs just use cheap non stick skillets they throw out when they are worn or scratched. Why become a materials engineer when you can just do the cheap and easy thing?
It sounds like the nonstick coating on your pan has been worn off.
I would recommend a new pan like this for eggs. You barely need any fat and the eggs won’t stick:
https://www.amazon.com/T-fal-Nonstick-Cookware-Thermo-Spot-Indicator/dp/B000GWK2X2?th=1&psc=1
I've been using a T-fal for several years with no problems. But the important thing to remember is that Teflon eventually wears out. You'll have to replace eventually. If you want something higher end, get a cast iron or carbon steel pan and keep it well maintained and seasoned.
Check out TheWirecutter's recommendations here. I've been using this T-Fal Professional Fry Pan for about a year and a half now, that was a previous recommendation of theirs and it's still the best pan I own.
If you have the means I'd swap it for this. https://www.amazon.com/dp/B000GWK2X2/ref=cm_sw_r_cp_apa_uOJlybTAW02QW
Or better cast iron. Teflon is inert but also a carcinogen.