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When I lived in Korea, a lady told me her mother had died a few days before. Being very new to the country and language, I told her "죄송합니다."
This works in English to say "I'm sorry" in such a situation. But in Korean, not so much. It makes it sound like you killed the person and are now apologizing.
What phrase or word can I use to express sympathy or regret over an unfortunate event I did not cause?
As you say, 미안 합니다 and 죄송합니다 aren't quite right, as they're more for taking responsibility for your own actions.
In my local culture (UK), we often respond by showing concern and asking some somewhat 'matter-of-fact' questions about the situation - e.g. "had he been ill for a long time?" (오래 편찮으셨습니까?), and this approach could be appropriate in Korean too. You could say something like "많이 놀라셨겠습니다" ("you must have been very shocked") if it were appropriate.
삼가 고인의 명복을 빕니다 - literally something like "I respectfully pray for the repose of the deceased's soul" is a phrase worth knowing, though it would be odd to come straight out with it on hearing the news.
For less serious unfortunate news, 유감입니다 is a common phrase meaning 'it's a pity', 'it's a shame' - though this would be too 'weak' as a response to something (like a death) that would have been very troubling to the person you are speaking to.
Your question is very difficult to answer because Koreans have very different ways of responding to such a bad news and it will entirely depend on sex of the responder, (social) position of the speaker and responder, age of the deceased, etc. Basically, there is no idiomatic expression such as "I am (so, very) sorry to hear that (the news)."
If a speaker is your friend, you could consider saying:
진짜! 마음이 많이 아프겠다. (Literally) Really? It must hurt your feeling a lot.
or
얼마나 마음 고생이 심했니. (Literally) How much you suffered mentally (emotionally).
If a speaker is your senior or elder as in your case, you could consider saying:
얼마나 상심이 크시겠어요. (Literally) How broken-hearted you are.
or
얼마나 마음 고생이 심하셨겠어요. (Literally) Honorific of 얼마나 마음 고생이 심했니.
'상심' literally means 'feeling of loss" and '마음 고생' literally means 'hardship (suffering) of mind'.
There could be more.
In Korea, they use
너무 상심(傷心)하지 마세요(마시고). -> Don't be too sad.
Or adding:
옥체를 보전(保存)하시기 바랍니다 -> Take good care of yourself (literally 'your body').
Here 옥체(玉體) is a honorific form of 몸.
