Note for the use of the words 미안합니다 + 죄송합니다: Sorry is an admittance of guilt in KR. Sometimes, english speakers use sorry to express that they are being empathetic towards a situation. But that's not the case in KR. So if someone's family member passes away, you don't say 미안합니다 or 죄송합니다. Unless you killed their family member :/
I believe 죄송 means apology, so 죄송 + 하다 = to do apology, if that helps people with the relative meaning.
Maybe this should be translated to something more formal considering the honorific.
I think it's for when you do a mistake, like if you broke your mom's flower pot.
"I apologize" seems more accurate, since the literal translation is apology-do. Avoids the mixup with the " I'm sorry" you'd typically use at funerals.