In your scenario, s/he would say "I feel nauseous. I think I have been poisoned."
However, out of the two phrases in your question, "I've been poisoned" would be the phrase a native English speaker would use.
To say "I have (I've) been" refers to the past tense. "Poisoned" is also past tense and suggests that someone else has already done the poisoning to you. (Which would be correct in this situation because you are using the same two tenses in the sentence).
To say "I am (I'm)" refers to the present, something that is happening now while "poisoned" is past tense, something that already happened. Mixing the two tenses (past and present) in the same sentence is what causes it to sound unnatural.