If I post an answer to a question that isn't a great answer in some way, and a downvote brings it to my attention, what should I do? My instinct would be to delete the old answer and post a new one instead of editing.
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No, by all means that is the last thing you should do.
Downvote means someone think the answer is wrong or not helpful. If you know what you done wrong and can fix it, improve the existing answer - that's the proper way of action.
Even if the downvoter already went away and won't come back to check and undo the downvote, editing bumps the question to the top, other users will see your improved answer and upvote. Usually you'll have +1/-1 meaning +8 points, and if your edit made it a good answer you might even get more.
Worst case you would lose some points - really not the end of the world.