My guess is that you need a new cassette.

New chain with skipping on one or more of an old cassette's sprockets commonly means teeth on those specific sprockets are worn. The sprockets on cassettes don't wear evenly - some are used more than others and wear sooner. Frankly, that's the only way I personally have figured out that a cassette is worn, if it doesn't work with a new chain in the way you're describing - I'm not good at figuring it out by looking at it - others may be.

The cassette might have well worked fine with the old chain because the wear on those sprockets matched the wear/elongation of the chain that was being used - but you don't know because it wasn't yours. The seller could have honestly said "it shifts fine". But now the worn teeth on those sprockets don't match up with the unworn links of the new chain