For a bike that old with cable housing in bad shape, it sounds like there's no question you need new housing. The housing is a critical part of the transmission. (For a system this simple, every part is critical.)

If one housing failed, the rest are probably close behind. Get new cables too. It can get hard to push old cable into a new housing, and they're not expensive.

Use the old housings as guides for cutting or buying new ones.

You may be surprised at how that project will improve your shifting.