Translation:I am sorry!
No. Both "I'm sorry" are used for apologizing for a fault. You could use it when you bump into someone but not to get someone's attention and not to express saddness for someone (eg. I'm sorry to hear that)
I've heard 사과합니다 used in k-dramas when someone has made a huge mistake. Is that more formal than 죄송합니다? Does it express a deeper regret?
Sagwa means apology (as well as apple) so I assume it means I apologize, rather than Im sorry?