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A Japanese grammar book will tell you that to make a 形容動詞(な adjective) into an
adverb, you add に after it. For example:

> **静かに** 歩いてください
>
> そういうのって **確かに非常に** 辛いな

However, recently, I’ve started to notice sentences where instead of に, で is
used. For example:

> **高速で** 車を動かしてください
>
> **早速で** 作ってみましょうね

This doesn’t make sense with me. It does not seem to be the で that marks
“method” either. For me, it clearly should have been

> **高速に** 車を動かしてください
>
> **早速に** 作ってみましょうね

Question: is the usage of で here grammatically incorrect? If not, then what’s
the explanation of it(why not に, etc.) ?

Note: this question is SPECIFICALLY about the use of に and で after 形容動詞. Not
just their general difference.
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question
grammar particles particle-に particle-で
When can 形容動詞 go with で?
> Question: is the usage of で here grammatically incorrect?

As far as 形容動詞 (な adjective) is concerned, yes.

That said,

> 高速で車を動かしてください

can be a grammatically correct sentence (if a bit an awkward one), if you see
高速 as a noun, short for 高速道路, and で as a location marker. That is different
from the adjective 高速な.

な-adjective's で-form, or continuative form (連用形), is used in different
context. For example, you use it when you connect two or more adjectives as in
高速で正確な作業 ("fast and precise work"). The examples given in the question are all
adverbial.

In general, many nouns and na-adjectives share some of the forms
superficially, but need to be parsed differently depending on the context. 自然
is another example that can be a noun and the stem of a na-adjective.
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answer