Further update: The service guy returned with a new blower motor, went
down to the basement, took one look at the burners that I had removed
for cleaning and said: "Ok, I'm condemning the whole unit, I should
have taken the burners out last time I was here and looked at them and
the heat exchanger." It looks like the warranty company will honor the
contract we have and will probably do so by paying us a cash amount to
replace the furnace. We'll likely have to add some of our own money to
upgrade that, since I suspect this furnace is less than 80% efficient
(again, most of the tags and model information has been painted over
and isn't readable).

Anyhow, thanks for the helpful replies and discussion on this. Just
thought I'd leave a feedback post, since it's been mentioned before
that most OP's don't bother to do that. Thanks again!