If those jockey wheels are worn, you should also check your chain and cogset, as they may too be worn and cause skipping. A worthwhile tool to have is that Park chain wear checker. It fits over a link and then the other end rests in another link, and if it goes all the way into the link, the chain is worn and should be replaced. Of course, if the cogs are also worn, they should be replaced along with the chain. The derailleur cable is another item that should be changed every year, and if the bike is subjected to a lot of harsh weather/dirt the cable housing should also be replaced.

Hopefully the cleaning you gave that pivot point will help the derailleur's upper pulley track under the cogs and tension the chain properly. I do most of the work on my bikes, and would never think to open that up myself.