Stuttering is complex! There's no ONE reason why an individual may stutter. Yes, anxiety and nerves play into it so if you're anxious, you'll probably stutter more. BUT stuttering can also be genetic or neurological, so you can stutter when you're alone and talking to yourself. There are a lot of theories as to why someone might not stutter when they talk in "other" voices. For instance, singing, choral responses (like when the teacher asks a question and the entire class answers at once), talking in an accent, sometimes even speaking another language can all be fluent speech for persons who stutter. Of course play around with your "deep" voice, but you'll likely return to your natural voice if you stop putting effort into it. AKA, meh, don't really try to stick to the deep voice.