It depends! "I am offended" sounds stronger and like you are stating a fact. "I feel offended" sounds softer, and like you are about give a reason why you feel that way.
"I am hurt" sounds like you have a physical injury if you do not explain more. "I feel hurt" sounds like you have emotional pain. "I feel hurt because you lied to me." "I am hurt and need a doctor."
Some of these uses are idiomatic, unfortunately.
You would never say "I feel be hurt."
If you want to use the present participle, it's usually after "like" and means you want to do something, or are describing your condition.
"I feel like crying because I feel sad."