I don't have DID so I am really hesitant to comment. But, I wanted to ask if you all have thought about seeking treatment (speech therapy) for the stutter? A speech language pathologist could help with it. If you live near a university with a master's in speech language pathology program they often have a campus clinic where you can get treatment at a reduced rated (or even for free sometimes).

I am a grad student and I am currently treating a client who developed a stutter following a traumatic life event. I really hope I am not overstepping... stuttering and other communication disorders are my field of study so I wanted to offer the resource in case you all decided you wanted to get treatment for the stutter at some point.