********** NOT A TEACHER **********
(1) I think that native speakers use both. It probably does
not matter which one you use, for native speakers would
understand -- especially in context (when you are expressing
your sympathy "for her").
(2) BUT I have to be honest:
A few people such as I feel that the "correct" version is:
I feel bad for her.
(3) A few very strict teachers remind us that badly is an adverb. Thus,
it modifies the way in which we physically feel with our hands:
I broke my fingers, so when I use my broken fingers to feel
something, of course I feel those things badly (in a bad manner). That is
a sentence that probably almost no one would ever say in his/her life.
(4) By the way, "feel bad" also means:
I did something of which I am ashamed. For example: I went to the
beach instead of my mother's birthday party. Now I feel bad about
doing that. As you can guess, many native speakers would use badly
in that sentence, too.
(5) Let's see what others say.
If I have given you any wrong ideas, I shall feel very bad.
P. S. I just remembered: I feel bad also means:
I am not feeling well. Yes, many native speakers use badly in that
P. P. S. Maybe if you are interested in "correct" English, just follow
this rule: Always say I feel bad to express sympathy, a guilty conscience,
or ill health. I do not think you will ever have an occasion to use I feel
badly (unless you happen to be a carpenter).
Student or Learner