adjective
BAD : I'm sorry for my terrible handwriting.
GOOD : I'm sorry about my terrible handwriting.
BAD : Unfortunately I can't come to your wedding because my exams begin on that day. I'm sorry for that.
GOOD : Unfortunately I can't come to your wedding because my exams begin on that day. I'm sorry about that.
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be sorry about + noun/pronoun: 'I'm terribly sorry about your camera - I didn't mean to drop it.' 'I'm so sorry about all the mistakes.'
be sorry about/for + doing sth : 'I'm sorry about/for taking so long to answer your letter.'