Burn-in likelihood has to do with the user and OEM, not just OLED materials.
From improvements in the efficiency of OLED materials to software developments and new testing techniques, OLED burn-in risk has been lowered. OLED monitors are generally a more sound investment than ever—at least for the right person.
CCFL-lit LCDs are so inefficient compared to modern LED-lit LCDs that you've probably spent enough more on electricity by now to have bought a more efficient monitor.
I can't speak to the environmental impact, though. Producing the new monitor emitted some amount of CO2, and powering each monitor takes some amount of CO2 per unit time. At some amount of use, the newer monitor will have lower lifetime CO2 generation than your old monitor.
Not OP but my electicity is <$0.10 / kWh because of where I live. It seems like it would take much more than 13 years to hit the break even point on upgrading the monitor just because of energy efficiency.
Even if the newer monitor has less of a lifetime environmental impact, throwing out the old still working one is still wasteful. It's already made and working. Using it longer lessens your environmental impact. If you repair the old one when it eventually breaks, that's still less of an impact than an extra ~20% electricity usage. Especially since electricity generation is getting greener all the time.
What about the degradation of the display? It’s ok because you can shovel it off to a poor person?
It’s very expensive (in terms of resources) non-durable consumer good. Which is a crazy waste even if you foist your junk off to some poor sap that is forced by circumstance to live off your table scraps.