If it was my bike, the first thing that I would do would be to have the derailleur hanger alignment checked. Whenever I've encountered a bicycle with rear shifting that didn't respond to normal cable tension tweaking, derailleur hanger alignment has usually been the culprit.
If hanger alignment turns out to be the problem, you're done and you've saved yourself $67.00. If the hanger alignment turns out to be okay, at least it's a common issue that you've ruled out.
__________________
My greatest fear is all of my kids standing around my coffin and talking about "how sensible" dad was.