It sounds like you have a worn out cassette.

When your chain was bad it was also wearing on the teeth of the cogs/chainrings. This changes the "spacing" between each tooth as well as the profile of the tooth (shark toothed). When you put on a new chain it won't fit the "redesigned" gear teeth any more.

You see one sign of this by the derailluer cage "jumping" as you put it. If you look closely at the chain as it wraps around the gear cluster you'll see it is lifting out of the tooth-pocket and riding on the tips of the teeth when the cage moves. When the chain can fit between the teeth again it falls into place, the cage jumps and the cycle starts over once more. This is called "skipping" and it can happen on both the rear cogs and the front chainwheels.