I think it IS a bit weird when you go about that sentence that way, but, try to look at it from this angle:
If we group the sentence’s meaning like this:
[素直で正直に][生きるようにして。]
It certainly feels very weird.
But, what’s actually happening is:
[素直で][正直に生きるように][して。]
(Frankly… / Honestly… / Please…) (So that you lead an honest life) (Do it)
By putting it all together, we get the final translation:
“Please, try to live honestly.”
And, to give a more concise answer for this, で is just a conjunctive here. It’s connecting the sentence rather than turning into an adverb, while 正直に is adjectivizing 生きる.
素直 has more of a “what I’m really thinking deep down” honesty nuance, while 正直 expresses more of a “right way/upstanding way of doing things” honesty nuance. So the “Please” at the beginning of the sentence has a “I’m begging you!” feeling to it.
Hope this helps, even if late!