Affidavits are seen as "powerful" documents by people in some countries - I just don't see why people need them in this process. The DS260 itself is a legal document. It is witnessed by the CO at the time you make an application, so it is, by itself, an "affidavit" (defined as "a written statement confirmed by oath or affirmation, for use as evidence in court."). So - an affidavit in these circumstances is pointless - in my opinion - but if it makes you feel better then it can't normally harm.
Regarding this specific scenario. The DV process does not depend on your parentage. One reason they ask for the long form (with both parents names) is that if you ever want to sponsor your parents to live in the USA, a paper trail has already been established. So - if you don't plan to bring your parents to live in the USA, that won't ever be an issue. If you do plan on sponsoring your parents the information in your DS260 will be "evidence" of what you claimed at the time of your immigration process. Given the lack of information on your birth cert, you may be challenged to prove your father is indeed your father. You can deal with that if and when that time comes. In the meantime, it doesn't matter.