I want to say: I speak very good Chinese. Can I also say 我很好地说汉语？ How to understand the differences in meaning?
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I don't think both of them have any grammar mistakes, Maybe just Chinese usage habit involved. Talking about someone's ability of language, most of Chinese will say "你/我汉语说的很好"。Some Chinese may say “你/我很会说汉语”， but I never heart someone say "我/你很好地说汉语". However, I think most of Chinese native speaker can fully understand "我很好地说汉语"， although it's a little weird.
很好地说汉语 simply sounds very stilted, it might just be something you need to accept.
Besides that, I think it's worth pointing out that adv+地+v usually describes one instance of the action, whereas v+得+adv can describe a norm or ability to perform the action in a specific way. For example, "他飞快地奔跑" is only describing that he is running fast now, while using "他跑得很快" can emphasise that he has the ability to run fast. When you say "I speak very good Chinese", you are emphasising on your general ability to speak good Chinese, rather than any particular instance of your doing so.