Chess is a great example you picture a 'move' as a new layout. Practicing physical activities falls under the same kind of thing where I can focus on where to place my foot or when to breathe etc even when not doing the activity.
If you want to get a sense for what non-linguistic thought would be, think more like your dog. The two language areas are the only functional difference between us and the animals. Once you start getting into the realm of things that dogs or other animals can't do, you're thinking linguistically.
My own personal anecdotal experience is that thinking in words is far, far slower than skipping them and sticking to concepts and connections.
Like I stated earlier, I don't think all cognition is linguistic in nature, but I think you'll definitely lose 'something' if you break your language part.
My argument is that without language, we'll be present-focused and unable to appreciate conceptual subtleties, due to the inability to name and remix thoughts. Like other nonverbal mammals.
I find this as well. I've often caught myself waiting for my inner monologue to finish narrating a thought while simultaneously being fully aware that I've finished the reasoning part and already know the answer. When this happens I actively try to let go of the narration and just take the results and run with them. It's difficult but sometimes doable, and (for me at least) there's about an order of magnitude speedup.
If true, I wonder how humans thought before language had evolved. Sounds like hell.