Key Difference: "I'm feeling sick" is about a temporary sensation. "I'm sick" is about being unwell, which could be short-term or more lasting.

They mean the same thing: you are ill, you feel ill. You can’t ’have sick’, you *are* sick or you can say *feel* sick. Sick is the term that describes all illness or sickness. But you can *have* a cold or some other specific sickness or condition: I have flu, I have covid, I have a headache, I have food poisoning.

I'm feeling sick - You are currently feeling sick and may get better very soon, within a few hours or maybe a day. I'm sick - You have been sick for an undisclosed period of time, the sickness could be incurable and you may have a terminal disease. I'm sick could also mean that you are perfectly physically healthy but that you have psychological problems, and that you are a danger to yourself and other people.

It is the difference between fact (being) and perception (feeling). Similarly, if you fall and hurt yourself, you can say what has happened "I am injured" or you can state what you perceive: "My leg hurts".

All correct. To not sound repetitive and to add more clarity, try to vary your pronouns and subject terms. When you say you are feeling sick, it means that you’re not feeling “well”. When you say that you are sick, that means that you are “ill”. Adding variety in your writing makes for better clarity and will keep your readers’ attention better.