Thank you! Yes, I HAVE unplugged it for now. I do not see any
corrosion on the plug, but there are tell tale signs on the plug
itself, like burn marks near the metallic part (the correct term
escapes me!), and the wire closer to the plug gets hot and soft as
well. Can replacing the plug be a SAFE solution, or will it still carry
some danger with it? How do I fix it if there is a near break in the
wires inside?

A properly replaced plug is safe. (Make sure you know which wire goes to
which prong; you can use an ohmmeter on the old plug and the cut-off wire to
verify this.) Simply cut off the wire far enough away from the original
plug to get past the break. Can you identify the exact spot that the heat
is coming from?