First, if you take the easiest route and repair the existing ceiling, the popcorn should cover any imperfections. That's probably why it's there in the first place. I don't have experience with popcorn ceilings, so that's for someone else to answer.
you could probably put up drywall over the existing, if you want; using long screws (2"?) and make sure you hit the joists - covers the popcorn fwiw -
Taking down existing and starting fresh with drywall will be messy but will produce the best result in my opinion.
Before anything, check to see if moisture was the culprit for previous repairs and take care of that.