That expression certainly isn’t a problem in French, where one would say ‘J’ai eu faim’ (literally: I had hunger). The use of ‘is’ in expressing the equivalent in English seems to just be part of our linguistic culture (or something— I’m groping for words here).
To go a little further: many counselors and therapists actually encourage their clients to consciously make the distinction between ‘is’ and ‘feel’ when expressing their inner states— a cognitive-behavioral approach known as ‘neurolinguistic programming’ (so fancy!).
A basic example: instead of ‘I am sad’ (which counter-productively suggests that sadness is both an inherent and persistent trait of oneself), the client is encouraged to say ‘I feel sad’ (which not only dissolves these attributions, but also stages this feeling for discussion).
[Edit: I just now saw that NLP is mentioned in the E-prime article linked by @anon29631895 @nemomeno. My right hand to FSM, I did not see this before writing my post.]
