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I understand that in these expressions the に isn't a particle (I think it
indicates the adverbial form of a na-adjective) but I'm not sure what で is:

> 必死 **に** 走る  
>  [_run like
> crazy_](http://ejje.weblio.jp/content/%E5%BF%85%E6%AD%BB%E3%81%AB%E8%B5%B0%E3%82%8B)  
>  必死 **で** 走る  
>  [_run like
> hell_](http://ejje.weblio.jp/content/%E5%BF%85%E6%AD%BB%E3%81%A7%E8%B5%B0%E3%82%8B)

According to [this answer](https://japanese.stackexchange.com/a/12713/19206)
it's ungrammatical to use で to make na-adjective adverbial. Still, the English
translation is almost the same but the one with で is stronger.
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grammar na-adjectives
Difference between 必死で走る and 必死に走る
Off the top of my head, it seems that nouns that originally modify a other
noun/NP tend to be "na-adjective" and always require に when modifying a
verb/verb phrase. 必死 is a noun on its own meaning "certain death" and is not
applied that rule.
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> According to [this
> answer](https://japanese.stackexchange.com/questions/12712/usage-of-
> particles-%E3%81%A7-and-%E3%81%AB-with-%E7%9A%84/12713#12713) it's
> ungrammatical to use で to make na-adjective adverbial.

If you look at it closely, Earthliŋ said in his answer:

> 的 makes 世界 into a 形容動詞 ("na-adjective"), which, when functioning as adverb,
> turns into ～的に.  
>  ～的では is simply ungrammatical.

The answer didn't say "～的で is simply ungrammatical", but it said "～的では" is
simply ungrammatical.

But this discovery doesn't seem to answer your question.

Similar to the word 必死, I found 元気. You can use 元気に走る、元気で走る、元気な姿 just like
必死に走る、必死で走る、必死な形相｛ぎょうそう｝ _look/expression_.

I think 必死で走る is made by omitting some expressions in the original phrase like
by omitting **な形相** in 必死 **な形相** で走る.  
This could be said 元気で走る is made by omitting **な姿** in 元気 **な姿** で走る.

If so, it is natural that you think 必死で走る seems stronger than 必死に走る, because
the latter phrase is a phrase as it is while the former one hides such the
expression like 形相 in it.
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「必死で走る」「必死に走る」 both sound perfectly fine to me, and I believe these are both
grammatically correct.

* * *

> I understand that in these expressions the に isn't a particle (I think it
> indicates the adverbial form of a na-adjective)

I think you're right. According to 明鏡国語辞典, 「必死」 is a 形容動詞/na-adjective, so it
conjugates to its 連用形/continuative form 「必死に」 to modify the 用言/conjugatable
word 「走る」.

* * *

> but I'm not sure what で is... it's ungrammatical to use で to make na-
> adjective adverbial.

In 明鏡, 「必死」 is also categorized as a noun. So the で here is a case
particle/格助詞, and is used in the sense of:

> で 〘格助詞〙  
>  ❹ 動作を行うときの **様態** を表す。「急ぎ足 **で** 歩く」...「笑顔 **で** 答える」「土足 **で** 上がる」  
>  （明鏡国語辞典）

* * *

> the English translation is almost the same but the one with で is stronger.

「必死に～」「必死で～」 don't really sound different to me... I wouldn't find it wrong if
the _crazy_ and _hell_ were swapped :D
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