Hello Apple,
There are two types of sentences in English:
1- Someone or something does something, and
2- Someone or something is or seems to be something
The verb in the first type of sentence is called an action verb. The verb in the second type of sentence is called a linking verb.
In the sentence " I felt sad" ,verb "felt" in the past simple tense is a linking verb.This linking verb reflects a "state of being". It is actually derived from the verb "to be," but the forms of "to be" are not the only verbs that can act as linking verbs.
Linking verbs has that name because they link the subject of the clause
( "I' in this example) with something that comes after the verb and either identifies or describes the subject, hence an adjective ("sad" in this example).When you say " I feel sad" you are expressing a state of being.
You should use the adjective ("sad" rather than the adverb ("sadly"), because the subject complement slot cannot be filled by an adverb.
"I feel sadly" would be inappropriate to use anyways, even if the verb "feel" was an action verb,which would mean perceive something by touching it. To say " i feel sadly " then would mean that you feel something like a fabric for example in a sad manner,which doesn't make sense !
So here "sad" is an adjective and you can't use the adverb"sadly".
If the adjective was " bad":
"I feel bad" would be describing my feeling as "bad" while using the adverb "badly" is possible and would give a completely different meaning.
" I feel badly" would mean I perform the action of examining the texture of an object in a bad manner.
I hope the difference is clear;)