I mull this over quite a lot. People of a certain generation and background (caveat here: I was taught to say them this way, too, but after doing all of my usual mulling, I got over it) tend to produce the following:
I am feeling/doing well.
where I would say:
I am feeling/doing good.
As I said, I've thought about this a lot (this is what I do in my spare time) over the years, and then recently, I heard someone say, several times, "I felt badly about that", and it caught my attention, because I think it's doing the same thing. Before I say why I have changed my ways, think about which of the following you prefer:
I am feeling sad.
I am feeling sadly.
I'm guessing it's the first one, right? Assuming I'm right (I can hear y'all nodding from here), that's because "feel" is what we might call a linking verb (and in the sentences above, "do" is functioning the same way) - that is, a verb that links the subject to the predicate (in this case, the subject is "I", and the predicate is what I'm predicating about "I" which is that I'm feeling sad). It creates an equals sign between "I" and "sad". "Sadly", by contrast, makes the verb "feel" here into an intransitive verb, because "sadly" is an adverb, which modifies the verb and tells us more about the manner of the feeling. In other words, a characteristic of the verbal action of feeling is sad. But this sentence is really saying that sad is a characteristic of me, not of the action of feeling.
So, looking again at the first sentences, and at the newest object of my grammatical mulling ("I felt badly") it's the same situation. In the case of "I'm feeling/doing well", it's a judgment of how good I am at feeling or doing, not a statement about my state of mind. In the case of "I felt badly", the adverb "badly" says something about the verb - I'm not so good at feeling, or I knock things off shelves when I'm feeling for them. But I would argue that we're really trying to say something about me, hence, I felt bad.