I had a friend do this. It was utter misery for over a year. Most of his teeth were shattered, so he had to wait for a lot of fragments to expel naturally.
Do not discount pain management that involves opioids. Not saying to use them for weeks nor do I know your personality for bad habits, but if it gets bad please don't suffer for no reason. Getting ibuprofen or acetaminophen with low dose codine may be a good middle ground and is even available over the counter in some countries. Extended pain is mentally exhausting and isn't worth the hit on your mental health.
f you are struggling taking pills, get liquid ibuprofen. Sometimes you can get a chemist to make a suspension for you, otherwise get childrens. I do that if my throat gets too infected and I am unable to swallow. Honestly, it works far better than the pills and I needed a lot less.
Best thing is to be honest with your doctors, even if you do not want anything stronger. Be sure to communicate any discomfort due to ill fitting dentures. Ask questions if anything unexpected comes up. My friend's doctor was super shitty and didn't even tell him about all of the left over fragments that still had to come out on their own.