Thanks for the replies, everyone, and apologies for being slow to follow up on this. It looks like he has an "oronasal fistula", so whenever he eats, the food/liquid goes up into his nasal passages, causing the intense itching/pain.
It looks something like this:
Over the last two days the problem has gotten much, much worse (hole has grown) and now this sweet dog growls and then bites me whenever I go anywhere near his mouth. I am unable to hand feed him now, and he won't eat on his own even though he cries all day that he is hungry. The situation is only correctable by surgery, but his gums/bones are too far gone to make that possible.
He is also unable to sleep from the pain-- used to sleep 20 hours a day, now sleeps barely at all.
I can't believe how quickly this happened. Quality of life is now bad enough that I am likely to ask that he be put down at our next vet appointment tomorrow night.
The worst part is that he doesn't look "ready to go". All my other elderly animals have made it known through their look or demeanor that they were ready to move on, but this one still has a lot of spunk left in him despite his overall frailty, so this is really eating at me. But he just lies on his bed, eyes wide open, obviously entirely mentally alert but staring at nothing, waiting for time to just pass, and doesn't want to be touched, so I know it's time.