FEEL is an action or state verb?

I know we can say either: How are you feeling today? I'm not feeling/I don't feel very well.

also: I saw a few times: I'm loving you instead of I love you.


FEEL can be either an action or a state verb.

I am feeling well today. (action - experiencing a particular feeling / emotion)
I feel our government is not doing enough for our country. (state - having an opinion)

I love you. (state - having a deep, tender feeling)
I am loving you. (action - embracing / caressing / etc...)
For simple exercises, as a substitute for "think" or "believe" use only the non-progresive form.
I feel we should go, I felt he was the best choice.

When you use the continuous form, you are emphasizing that it is very current, and probably temporary. I'm feeling sick - it's immediate, it's right now, it's going to end. I am feeling that = I am sensing = I am picking up on the idea that... it's time to go, you don't like my husband, your feelings are hurt, whatever.

The only way "I am feeling we should go home now" works is if something has changed, quite recently, and it has changed the atmosphere. For example, you are at the house of friends, but suddenly your host and hostess are arguing with each other in a way that makes you feel very uncomfortable. You may be feeling that it's time to go. It's not a belief, it's a feeling! This is pretty complicated for an English-learning situation, so use "feel" in your example.
Students: We have free audio pronunciation exercises.

I was doing an exercise and the instructions were as follows: Choose the correct form of the verb depending on whether in this meaning it is an action or a state verb. If both the Present Continuous and the Present Simple tenses are possible, use the Present Continuous tense.

There were such two sentences:

a) I am feeling we should go home now.
b) I feel we should go home now.

And according to the answer key only answer b is correct. If the word "feel" is both an action and a state verb then the answer a should be correct. Is that right?

 BarbaraPA's reply was promoted to an answer.
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Thank you for your clarifications!

So why do we say "that feels good" instead of "that's feeling good?"