My old chain was skipping a lot, so I replaced it with a new one. 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.
The cassette does not look worn out. The rear derailleur seems to be indexed correctly but curls up very tight in highest gears.
I cut the new chain to be the same length as the old. In order to do this I only had to remove one link in the chain, which makes me wonder if the previous chain was cut too short (or too long).