You are on the right track. Purchase premixed topping or general purpose compound. If any surfaces have gloss paint, you must de-gloss or sand first. Use sanding screen and a block to get the surface you want. Apply any compound in thin coats. Defects can be hidden by texture.
You might look into hiring a drywall contractor to do these repairs. They are surprisingly reasonable in price and will tell you all about the clowns that originally did the job.