According to Cambridge Dictionary:
but (conjunction): used to introduce an added statement, usually something that is different from what you have said before.
I'm sorry, but I think you're wrong when you say she did it deliberately.
The example given is quite similar to yours, and both phrases are and sound perfectly normal. I'm awfully sorry, but I haven't done my homework/assignment. sounds like the person is apologizing to whoever asked him/her about it, and letting him/her know that the homework/assignment was not done.
One workaround for your phrase could be I'm awfully sorry for not having done my homework/assignment., but be careful if you do choose to use this alternate phrase, because the context is different:
I'm awfully sorry, but I haven't done my homework/assignment.
The person who asked about the homework does not know it hasn't been done, it's unknown.
I'm awfully sorry for not having done my homework/assignment.
Either the person who asked about the homework knows it hasn't been done, and is scolding (for example) him/her, or it has been stated before.