So you should know that in 'I'm feeling well' Or 'I'm well' the word 'well' is used as an adjective (NOT adverb) to mean 'in good health', the opposite of 'ill':
'I don’t feel very well.'
'Get well soon!'
'How are you today?' 'I'm very well, thanks'
However, it can also be used as an adverb, as shown in the examples above, to mean 'in a good, right or acceptable way'.
'Your English is good.' but 'You speak English well.'
'It is a good song and the singer sang it well'
=An adjective (good) describes a noun (song).
An adverb (well) describes a verb (sang).
'Good' is an adjective, and 'well' is its adverb. The opposites are 'bad' and 'badly'.
However, 'well' can also be used as an adjective meaning 'in good health', the opposite of 'ill'.
Also, 'good' can be used as a general reply to a greeting to mean 'healthy':
'How are you?' 'I'm good.'
'I don't feel too good today.'