Plus, who can ever forget that famous James Brown soul-music hit-tune -- "I Feel Well"?
Whatttttt??? Do you mean to tell me he wasn't singing about having sensitive nerve endings in his fingers??
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More seriously, though, "I feel good" is a natural change in the language. Setting the issue of predicate adjectives and simply focusing on meaning, "I feel well" is not the same as "I feel good"; i.e., the difference isn't simply an improper use of an adverb as a predicate adjective. Further, there really isn't an efficient alternate that captures the meaning; "I feel happy" might be close, but it's still not the same as "I feel good".
There is a void in the language, and "I feel good" is filling it.
Moreover, since "good" is an adjective as well as an adverb, it's not completely incorrect to use it as a predicate adjective. The problem really comes because "good" is an irregular adverb. Normally in English when an adjective is used as an adverb, a "-ly" suffix is added. "Slow" becomes "slowly". "Glad" becomes "gladly", etc. But we don't change "good" to "goodly" when we use "good" as an adverb. (In fact "goodly" is an adjective.) If "good" followed ordinary rules, James Brown would be perfectly fine singing, "I feel good", while a person with a well-developed sense of touch would say "I feel goodly".
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"Slow" and "slowly" is one of the incorrect uses that bugs me. It's not "Drive Slow"; it's "Drive Slowly".