It's an older DeLonghi. We have several, but this is the only one that
is going bad like this. I replaced the plug housing with a modular one
from Ace, but the cord itself heats up. Do I need a better plug, or do I
need to open the appliance housing itself (if possible) and replace the
entire cord? Other than the heat, the cord does not seem brittle or
frayed, nor was the cord in a location subjected to movement or abuse.

Along the same lines, one that was taken to work had its plug replaced
by the building maintenance with one claimed to be "heavy duty", but I'm
not sure what is meant by that -- and it's a grounded plug, while the
cord is just plain polarized (meaning the ground plug is useless).
Replace with a proper two-wire plug?