Should I Avoid Sounding Pedantic?

One of the questions I fear the most in the workplace is the seemingly innocuous "How are you?" I never know how to answer this question without sounding either pedantic or stupid.

When I first started learning English at 7 1/2 years old, the first grammatical rule that got drilled into me was the distinction between "good" and "well." By the time I moved to the United States, I cringed at hearing anyone say "I'm good" or answer the dreaded question with "good." However, during the cruel years of middle school and high school, I did everything I could to fit in. Which meant inserting at least two "likes" into every sentence and adopting "I'm good" into my lexicon. This continued to work for me throughout college and I stopped experiencing knots in my stomach every time I heard a grammatically incorrect use of "good."

However, this problem is now recurring in the workplace. A typical scenario goes something like this:

1. A coworker asks "How are you?"
   
2. Mini anxiety attack. Think, Irina, think.
   
3. Option 1: respond with "I'm good." How painful. That is so grammatically incorrect. I cannot do this. What if the coworker thinks that I believe this to be the grammatically correct way to respond? Then he will judge me for getting such a basic rule wrong.
   
4. Option 2: respond with "I'm well." That sounds way too pedantic. What if the coworker thinks that I consider myself than everyone else in my quest to use correct grammar at all times? Especially since he himsef probably have used "I'm good" many times before.
5. Option 3: respond with "I'm doing well." That sounds a little bit less pedantic than option 2, but it is slightly long-winded. What if the coworker has to run? What if I have to run? Will he think I am dragging out this insignificant exchange? What if I stop over analyzing everything? I really should answer now. This 2-second pause is becoming really awkward.
6. Final answer: "I'm good. How are you? Does it look like you will be having a busy day?" Oh good, I said a few things after the "good" to distract him and divert his attention to his own thoughts. This way, he will not have time to evaluate I answer.

Maybe I am over analyzing, but I go through this thought process (in lightning form, obviously) whenever I am asked that question. What do people think? Does it matter at all? Should I just suck it up and answer with "well"?

UPDATE:

As readers on Brazen Careerist have pointed out to me, supported by Grammar Girl, saying "I'm good" is a perfectly grammatically correct response. In this instance, "am" serves as a linking verb and it is grammatically correct to use adjectives after linking verbs. Therefore, "good" is an adjective modifying the noun "I." The reason I was confused before is that I was perceiving "good" to be the adverb modifying the verb "am." But it is not. It is an adjective. I hope this makes sense.

Also note that, per Grammar Girl, it is actually not entirely correct to respond with "I'm well" when asked how you are doing. "I'm well" refers to the state of your health, not to the state of your happiness. So you can use "I'm well" to mean that you are now feeling better (than you were before...presumably you were sick), but not to mean that you are in a good mood, etc.