In relation to thinking and doing without verbalising, there's one activity where my brain has clearly switched from one way of operating to another.
As a teenager in school I learnt to touchtype. I don't know if you have ever learnt this, but basically you have a blank keyboard and your fingers have to find their place and stay in place in order to find all the keys around them. It was extremely hard and slow going at first - not to mention very repetitive - but then a breakthrough moment came when one could keep one's eyes on the copy page all the time, and every finger knew where it should go. After that, it was just a matter of picking up speed. Very useful it was, and all in Spanish, of course.
Fast forward a couple of years and I began to study English at uni. We had projects to submit to different classes, typewritten, but that didn't worry me, consummate typist that I thought I was. Hah! The first time I tried to touchtype in English, I almost fainted! I thought I'd lost it completely! You see... I was loooking at my copy page, my fingers were typing away, and there on my sheet of paper was some kind of phonetic transcription of what I was trying to write. This kept happening, and after the initial dismay I realised I must be "hearing" the words in my head as I read them, and then typing their sound. This hadn't come through in Spanish, which is written basically just as it sounds, but here was proof - in some kind of pidgin - that my typing was linguistically mediated.
However, I had no way out other than to type the dratted papers in any case, so I stuck with it and over a few weeks it seems that I was able to train my brain to recognise not sounds but graphs (pictures, in some way), which is what I do now, touchtyping in English, Spanish or French.
In fact, I don't even know whether my brain is linguistically aware of what I'm typing -- it sounds more like the "from the eye to the hand" mechanism that someone mentioned earlier on this thread.