Feel well, I think, is used only to refer to a good
health condition. Feel good, on the other hand, can
also be used in this way, but is also very commonly used to refer to
a good emotional condition.
Both answers are correct.
Feel well and be well refer to one’s
health. Well in this case is an adjective. This is
the only use of well as an adjective—to mean healthy,
Feel good means to feel happy.
So in Bocchi‘s answer above?quot;I can't feel well but I feel good
this morning"—the sick speaker in the hospital is saying that he
can’t feel healthy, but he does feel happy, or in a good state of mind.
Compare feel good with feel bad
(click here). Feel good means
to feel happy; feel bad means to feel unhappy.
(Feel badly, as well as feel bad,
meaning to feel unhappy, guilty, or uneasy, does exist, as described
in Quirk et al., A Comprehensive Grammar of the English Language
(Longman, 1985, p. 408?09). It has become acceptable, according to The
American Heritage Dictionary's usage entry under bad.
Still, Quirk notes that ?there are prescriptive objections to the
adverb form?em>badly with feel.?