I would just pull everything apart and clean it all. The biggest thing beyond the shifting problem is what is underneath the nasty gross disgusting tape are the bars underneath in good shape or has the previous owners soaked sweat started corrosion, I would never want to leave a used bike with used bar tape it is gross and it is everyone unless someone is super meticulous about cleaning and sanitizing after each ride which is a small portion of riders. Plus with all of that out of the way you can replace cables and housing and flush everything out and just have a better functioning bike.

If you aren't good with mechanics you can always ask your local shop to take care of it. Now is the perfect time when winter is upon us and they are slower so you will get the bike back faster generally.