There are different schools of thought here. Some would tell you to replace the cassette in the condition you describe. I'm of the replace no cassette before its time school. IMO the only reliable way if knowing if it's time is to install the new chain and see how it goes. If the chain runs OK, then you're good to go.

If it skips under load, the cassette is toast. As the cassette nears the end, a new chain will run rough (you feel it in the pedals especially under moderate load. Here you can make a jusdgement call as to how much more you want to squeeze out of it. For example, on my commuter I'll put up with anything short of slipping, but on my road bike, 100miles pedaling with a rough train is unacceptable and I'd replace it.

BTW- you are definitely getting sub pat performance out of your chain lube, or more likely how you're applying it. Unless you're an animal riding in the Alps, you should do vastly better on a chain, something like 3-5,000 miles service life, with 10-20,000+ miles service life on a cassette.