Simply draining may be insufficient for basements; what provides better conditions is insulating the outside of the basement wall. If you then also choose materials that do not require a waterproof layer on the outside of the outer wall, it becomes even better. If you don't have any of that, you should avoid materials on the inside of the wall that create a tight layer, such as what you described exists in your basement. To remove the moisture, you should remove the fiberglass wallpaper. We have a basement that was redrained and got Podrän as insulation on the outside, no moisture problems with the same conditions as in your basement, both with painted walls and painted fiberglass wallpaper.