It depends whether you are using British or American English.
In British English, you'd say 'I'm ill'. This means 'not well'. For speakers of British English, 'feeling sick' means feeling nauseous, and 'being sick' means vomiting.
American English uses 'sick' to mean 'feeling unwell' in a more general sense, and reserves the word 'ill' for more serious and long-term conditions. Many people use the terms interchangeably, however. This usage has also been adopted by some younger people in the UK in recent years.