I used to from time to time imagine my half of a conversation in which I was showing around someone notable who had traveled through time to get to the present day. Maybe someone from 1,000 years ago or maybe from 50. That is, for entertainment, not to cope with anything. I never felt like my imagination was quite good enough to turn it into fiction.
In general though, I don't have a monologue in a continuous sense. I frequently imagine saying things, imagine other people saying things, occasionally imagine saying something to myself, but I would never say that's how I think exclusively. When I am having trouble with a concept or problem though, I tend to return to verbal analysis - a narrative or verbal description helps me figure out things that I otherwise struggle with.
If I am writing, I might be hearing the words in my mind, or I might not. If not, I might reread what I wrote and then feel like editing it, probably because I wasn't conscious enough of how it sounded. So, really, I don't exactly relate to having or not having an "internal monologue". Thinking one type of thought all the time seems weird to me.
As far as this post goes, I didn't know what it would look like until I was done, so I'm not necessarily conscious of how I organize things at all.