I've known two bad cases of schizophrenia (who could not hold onto a job), the negativity comes from the effect it can have on the person, not the word itself. Unfortunately the bottom line in society is that if your mind behaves "strangely" then others will shun you, if your body behaves strangely (say, cancer) then others will act with compassion, provided it's not contagious (eg:leprosy). On the bright side I think there is a lot more compassion and understanding toward "thought disorders" today than when I grew up in the 60's.
Changing the name is at best temporary relief from the stigma, the new term will eventually pick up the same stigma as the old one. As a "functioning schizophrenic" you have the opportunity to change people's attitudes by example, I understand it takes a great deal of courage to do that but you will never change anyone's attitude by "staying in the closet". For example when I think of Stephen Fry, I think educated, witty, curios, homosexual, intelligent, atheist, bipolar, introspective, honest and open, which when all rolled together creates an interesting and likeable personality.