I'm with Frank on this. "Be" verbs take an adjective, so "I am good" is correct. Many people think "I am well" sounds more "educated" or "elegant," but it means "I am no longer sick." You can always just say "I am fine," since people who don't understand grammar have yet to start "correcting" that to "I am finely."
Here is the good vs. well discussion from merriam-webster.com:
An old notion that it is wrong to say “I feel good” in reference to health still occasionally appears in print. The origins of this notion are obscure, but they seem to combine someone's idea that good should be reserved to describe virtue and uncertainty about whether an adverb or an adjective should follow feel. Today nearly everyone agrees that both good and well can be predicate adjectives after feel. Both are used to express good health, but good may connote good spirits in addition to good health.
Although this discussion uses "feel," "be" ("am," "are," "is," et al) and "feel" and other verbs are linking verbs. Linking verbs take adjectives.