As others have said the only difference in 9s and 8s chains is external width. Since both the RD and sprockets engage the rollers inside the chain, the outside width isn't a factor at all. If you were reporting a problem with the chain dropping between the front chainrings, that would be a width problem, but you're slipping problem cannot be related.

leave the chain alone, and replace the cassette and you'll be fine. If you can't find a decent cassette, definitely go with a new 9s bike, but since that will probably come with a chain you can keep this one as a spare.

FYI- monitor chain wear in the future, and you'll be able to replace chains before the cassette is toast. Most of us get 3-5 chains in before having to replace the cassette. For more info relating to how to monitor wear, search here under "chain stretch" or google the same term. There's lots of opinions but the general idea is the same.