Strictly speaking, 'well' (the adverb) is more correct, but nowadays either can be used in this sense. It's also worth noting that 'good' has a much wider meaning when used with 'feel'. 'I don't feel well' can only refer to your state of physical health. 'Good' can also describe your emotional state, especially in expressions like 'I feel good/don't feel good about ....', in the sense of happy, satisfied or confident.

Both sentences are correct. I would normally say "I don't feel well" but either way, people will understand what you are trying to say.

Hello Cherry: Either sentence is adequate. There is no critical rule about communicating that you do not feel well.

Both are fine although quite formal. Many people might say "I don't feel very well".

I don't see which one is wrong. People could use them both.