Despite what others claim, I don't see anything that looks worn.. You seem to indicate skipping is the main issue. What do you mean by skip and when does it occur? Skipping as I imagine it is only when the fully engaged chain actually roll over the teeth of the sprocket. This generally will only be a new chain on worn out sprockets and only when a lot of power is applied to accelerate or on a very hard climb.

If by skipping, you mean that the chain isn't fully engaged with the sprockets and it's side plates are riding on the tops of the teeth, then that is likely to be something other than wear.

Is this friction shift or indexed shift? Indexing might need some adjustment of the barrel adjuster. Either on the RD or somewhere else. Downtube or on the shift levers themselves if they are a shift lever only and mounted on the bars.

It's entirely possible that your RD hanger is bent and just coincidentally about the time you changed the chain. So the new chain shows the issues of a bent RD hanger more so than your old chain. Sometime a little hard to see twist of the hanger can make a lot of problems.