What do we do about abandoned questions, with a correct answer that was never accepted?
If we suspect the original poster forgot about it completely?
Not sure if we have any sort of process, or if we just let them say that way?
Leave them. It's nice if the asker marks the answer as accepted, but not really required; without an accepted answer the answers are just sorted purely by votes, so the community can upvote the right answer to make it show up first in the list. The system generally considers a question "unanswered" if it has no upvoted answers, not if it has no accepted answer; for example, this is how the unanswered list works
If a question didn't got accepted, it might have a reason. And it might take much time, to make a very good decision without the asking person.
I often receive bonuses for old questions which get accepted lately - I guess, because people ask again a question, and get told to work on their accepting rate, so they visit old questions and vote the best ones up.
Spending much time on old questions doesn't seem rational for me.