The wall that wobbled in the bedroom had 2×4’s that were rotted. That meant replacing them. With that done, we insulated the wall, replaced the window, and now the room is cozy warm and no wobbly wall.
All this stuff was fixed before we purchased the home and was done before we moved in. It is now only needing flooring, paint and the rest of the inside repair. But it is so worth all we have had done to it before moving in. It was even debated if we should just not do it at all. But with just the minor stuff now left to us to do, I think it will be worth it.