Unfortunately there are a couple issues with the premise. I'll try to address those, and then if that changes how you phrase the question maybe I can address that.

I suspect that the use of characters has played a very large role in the way that the Chinese think. Is this accurate? How?

This is problematic because it's drawing a causal relationship between a script and thought. This can quickly get into the rocky territory of strong Sapir-Whorfism.

In fact there's evidence that it doesn't have the sort of impact you might be imagining. As a simple example, you have 他 她 它 牠 祂 all being third-person pronouns, all pronounced tā, meaning "he" "she" "it (inanimate"" "it (animate)" "He (God)". If we were actually thinking in characters, you would not see the mix-up of "he" and "she" in English by second-language speakers from a Mandarin first-language background. But actually that mixup happens all the time, because we're not thinking in characters when we speak Chinese. We're thinking in spoken language. (Note that there's room for debate about how humans actually think blah blah blah but at least for the sake of addressing characters we can leave it at that)

For example, the relative unity of China throughout the centuries in spite of a number of dialects was helped, at least to my knowledge, by characters allowing for written communication throughout the empire.

This is problematic for a couple reasons.

First, I'd argue that unity is overrated. There's been plenty of disunity in China.

Second, while there is no scientific distinction to be drawn between what's a language and what's a dialect, it's somewhat dismissive to refer to what was happening in China as "dialects". In truth it was a wide range of languages, some related, some not. Koreanic varieties, Japonic varieties, Vietic varieties and Sinitic varieties were all written with Characters. Then there's Tangut and Jurchen which while not using the same characters that the others used, there were other sorts of characters that were developed for these languages.

But even if we just look at Sinitic, there were a huge range of mutually unintelligible varieties being used.

The thing isn't that there was a unified writing system. It's that there was a written lingua franca. In other words, it's not that everyone wrote their own language the same way. Rather, they all just wrote another language but all wrote the same one, even though it often didn't reflect how they spoke. You can imagine it as similar to Europeans writing in Latin well after the fall of Rome, well into the more-or-less modern era (however you want to define that).

Now, that's not to say that there's not room for assessing the impact of written language on speech and therefore thought. I think /u/rusoved can possibly address this in some detail. However the current high-quality literature on that particular situation (to my knowledge) isn't that substantial at this point. But again maybe /u/rusoved can shed some light.

I remember you asked this a little bit ago but didn't get an adequate answer. I'd meant to write one up but didn't have much time. This still isn't as much time as I'd like to spend on this but maybe it'll be a reasonable start.