Can I say: "I feel poorly" or do I have to say: "I feel poor" for "I feel ill"? Or is it more correct to say:"I am poorly"?
Using poorly as an adjective means "unwell" or "somewhat ill," and you can certainly say "I feel poorly" to mean the same thing as "I feel ill."
"Poor" is an adjective with several different meanings, and if you said "I feel poor" it would most likely be taken to mean "I feel like I do not have enough money."
Saying "I am poorly" is not more or less correct, but it changes the meaning of the sentence. The difference between "I am poorly" and "I feel poorly" is the same as between "I am ill" and "I feel ill." It's possible to feel poorly without actually being ill, and vice versa.