Wallpaper and Wallcoverings - Bathroom wall & ceiling problems

About a year ago my wife attempted to remove wallpaper that was in one of our bathrooms. When she removed the 4" border around the top she could not get the paper backing off and now there is a fuzzy surface where the border used to be. On top of that I also have numerous spots on the walls and ceilings where the paint is blistering. This most obviously is due to the lack of an exhaust fan in the bathroom.
I will be installing a fan but I am wondering if I would be better off just replacing the sheetrock or should I try to clean up what I have there. The bathroom is rather small 6' x 8' so it would not be a lot of sheetrock.
If I do try to repair what I have is there any suggestions on what to use to remove the fuzzy border backing?
Thanks

Replacing the drywall would be too much work!

Was the room repainted after the wallpaper was partly stripped? Leftover wallpaper adhesive can also cause issues with the paint.

The best plan of attack would probably be to sand/scrape what you can and then coat the walls with an oil base primer. That should give you a suitable surface to skim coat, sand, prime and paint.

Please post pics if you can -- I'm following how this type of solution works. Thanks!