I'd agree that "I'm well" has come to sound stilted in many situations, and that the easy way to avoid the problem is to use "I'm fine" instead, which sounds fine both to old purists such as myself and to those comfortable with the current usage.I prefer "Fine, thanks" or "I'm fine, thanks. How about you"? "I'm well, though correct, sounds stilted to me.
Still, in the right context, an "I'm well" will increase the regard in which you're held, just as it will sound stilted in the wrong context. I'm always insanely impressed when I hear someone under 30 use "I'm well."