When I think about it, your situation is not a common one for me as a native speaker. If I said, "I've done my homework", it's because my attention was on the present - perhaps the teacher wanted to know if the homework was ready to hand in at the time of speaking.
I would probably say "I did my homework" in a different context, perhaps in response to, "When did you do your homework?" The attention is on the past.
People may say sometimes, "I've done my homework; in fact, I did it yesterday." This adds reassurance and emphasises that you really have done it. In this situation, the extra words wouldn't seem verbose to me.