Interesting hypothesis.
I don't think I do, and my assumption is that most other people don't either?
I learnt to speak well before I learnt to read and write. The process of speaking without any knowledge of the letters was deeply embedded in my brain.
People that are unable as adults to read or write, or have severe impediments to reading/writing in the same way that I do (e.g. dyslexia) don't speak any differently.
If I try to, I can picture how my spoken word is written down, but I'm pretty sure when I'm talking in a more "autonomic" fashion I'm not really picturing the words?