How about "I am unwell today." Since the verb is really to be, it seems like it would/should be acceptable.
I second that. I would translate that as "I feel bad today".
Isn't "I feel unwell" the same thing as 'I do not feel well"?
Why is "ill" not accepted?
Ill is more like a sickness not a temporary feeling
These are roughly synonyms.
Astszi mi-e rău (mai corect)
Can I say 'Eu sunt rău astăzi'? Would that mean I am a 'bad person'?
Its not really correct. It doesn't make sense. It would mean roughly what you said it would but its that kind of sentence that cannot be translated "understandably". You can also say "Mi-e greață astăzi" or " Nu mi-e bine astăzi"
"I am feeling badly today" should be accepted.