While you certainly can feel a condition like claustrophobia, English writers overwhelmingly prefer the adjectival form to express the feeling. The same is true for other conditions like fever.
Rarest – feel fever:
She could feel fever burning in her mouth and forehead, her hands dry and throbbing on the wheel.
Rare – feel a fever:
Take 1 teaspoon of this mixture when you feel a fever coming on.
Common – feel feverish:
About three days after the “accident,” I continued to feel feverish.
Often, there's a difference in meaning between the adjectival and noun form. For example, feeling wet is very different from feeling water. That's because feeling an adjective tells you about the sensation, whereas feeling a noun tells you about the source or cause of the sensation (and may imply a judgment or conclusion about it).