Not caring about it actually dwarfs the anxiety that comes with stuttering so you can speak more fluent and also think/react better in the situation. But you should psychologically figure out how to do it. I've had a speech disorder since I started to talk when I was only a couple years old but I didn't care about it and it was totally not important to me to stutter for many years until I was around 15. Idk what happened but a day when I was at school in literature class I suddenly started to think what if I stutter during reading out loud a poem for the class and I would be ashamed in front everyone. Since that day I'm super self conscious about stuttering before every word I say to someone. And it kind of have a negative effect that the more I try to hide my stutter and to not stutter in front of someone the more stutter I do. and there is a great psychologically impact of that on me with every stutter I do in this condition (you know shame, anger, self hatred, etc). But I know that if I stop to be so obsessed about it I can kill this monster inside of me. Since my stutter is mild and I can speak fluently when I'm not under pressure and when I'm alone. In near future I definitely finish this ridiculous circus once for all.