Chain keeps skipping - did I cut it too short?

The cassette does not look worn out

A cassette that is worn does not look worn.

It'll skip anyway despite not looking worn.

I suspect your cassette is simply too worn.

(Interestingly, for chainrings the opposite is true: a chainring can look very visibly worn due to the non-even wearing at different crank orientations, and still work.)

The new chain barely skips at all, except for in the highest gears/smallest three cogs on the back cassette. The skipping comes progressively often as the gear increases.

This is exactly the symptoms of a worn out cassette. The skipping happens at the smallest sprockets. The smaller the sprocket is, the more likely it's to skip.

The skipping always happens when putting a new chain. With the old worn out chain, it won't skip unless it's practically destroyed as the cassette wear matches the chain wear. After a new chain is put in, the wears no longer match and thus the skipping starts.

When you are on the largest chainring and largest sprocket the rear derailleur should look very extended. Some movement should still be possible. It should shift freely from second largest to largest sprocket and you should be able to e.g. push the chain to the chainstay.

On the small-small combination the derailleur shouldn’t be completely folded, i.e. there should still be some spring to it. There absolutely shouldn’t be any slack i.e. the chain visibly hanging down.

Generally the chain should be as short as possible to improve shifting performance and avoid chain bounce.

You don’t need much chain tension. If it skips even on the big chainring your cassette is probably worn too much.

The skipping may come progressively often because you apply progressively more power (higher gears). I assume you are trying the smaller cogs at similar speeds.

Chain length may be an issue, but most likely the issue is the cassette. If your old chain skipped a lot, it is almost sure that it abraded a lot of of material from the cassette, especially when applying higher torque (---> smaller cogs on the rear are more worn out).