In "I feel strongly," strongly
is an adverb modifying feel. This use of feel means "to
be conscious of a specified kind or quality of physical, mental, or
emotional state: felt warm and content; feels strongly about the
election."* A synonym of strongly here would be intensely.
The correct and logical way, as you note, to express that you feel
sad or disturbed is to say "I feel bad," not "badly." In this
feel is a copulative, or linking, verb similar to be.
These verbs are followed by adjectives, not adverbs. Other common
like be and feel are appear, become, get, look, seem,
smell, sound, taste, etc.:
The weather was terrible yesterday.
I saw John. He appeared confused.
Mary worked too hard and got sick.
You look happy today!
Your answer seems correct. Let's check it out.
Mm! It smells delicious! What is it?
That sounds good. I'll see you at 6 then.
I'm really sorry. I feel very bad about saying that.
Please accept my apology.
If you say "I feel badly,"
it would mean, literally, that you have little normal feeling in the
nerve endings of your fingers, and so your fingers can't properly perform
their function of feeling. It would be like "I run too slowly/ badly
to be in the race," or "I can't hear well - I hear badly - please speak
louder." Badly in this case would be an adverb. The sentence "I feel badly," meant
in this sense, would be very rarely spoken.
You sometimes hear "I feel badly" instead of "I feel bad." The speaker
may be trying to be very correct or may think that "bad" is close in
meaning to "evil" or "wicked." Whatever the case, "I feel bad"
means "I feel sad / disturbed / unhappy / disappointed/ guilty." "I feel strongly" means "I
*The American Heritage Dictionary of the English Language. Houghton
Mifflin Company, 1996.