Is に turning these nouns into adverbs?
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None of your examples are being used as adverbs.
Your first example is the fault of lack of comma usage. Read this way:
「しかし、期待を胸に、中学校に赴任すると、理想と現実は大きくかけ離れていました。」 But, with my hope in my heart, I transferred to junior-high, whereupon my dreams and reality drifted apart dramatically. - - - - 胸に = in my heart
The other two examples are the same as far as using 手に (in hand), with one of them simply not containing a comma.
Kato took the important letter in hand, and headed again for Panama.
「彼女は手紙を手に、階下へ降りてきた。」 She took the letter in hand, and came down to the hallway.
Rules for use of commas are not quite as strict as in English. Usage also varies by writer. Often, where we would expect to see a break in the sentence there is none. Learning how to parse these types of sentences takes time.
If you do want to change nouns into adjectives or adverbs, use ～的 or ～的に as detailed here.
Also recommend that you get used to using「」 for open/close quotes in Japanese.