In English, the verb to feel is often used as a stative verb and in general, stative verbs do not take either continuous or progressive forms (and contrast with dynamic verbs which can).*
"I feel relieved." = "I am relieved."
To emphasize an awareness, consideration, or contemplation of the state (i.e., that act of feeling), one might use feel as a dynamic verb.
"I am feeling relieved." = "I am aware of/thinking about being in the process of relief."
However, this will still feel marked to some native speakers without something else to draw attention to a change in state or a temporary nature.
"I am feeling relieved, now." = "I was not relieved, but that has changed/is changing."
*The Farlex Grammar Book has a nice summary on stative verbs which includes this,
However, some stative verbs can be used in a continuous tense in certain situations, as when describing a temporary state that has begun and will end. This is becoming more common in modern English, and the prescriptive rule that stative verbs can never be continuous is becoming less strict.…