According to this answer 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.