Erratic Rear Derailleur Behavior
I replaced a Shimano 11x24 cogset with a 12/28 cogset (12/14/16/18/21/24/28). If I put enough tension in the cable to move from 12 to 14, the next click causes the chain to shift from 14 to 18. (Similarly, in shifting outward, the chain skips from 18 to 14.) Decreasing the cable tension removes the skipping phenomenon, but then I can't shift from 12 to 14.
I thought maybe the chain was moving too far outward when shifting onto the 12 gear (thereby requiring too much cable tension to move it back inward onto the 14 gear). But I've set the limit screw so that the outward movement is as little as possible.
Here's something that may be relevant. Both cogsets have seven cogs. But when putting on the new cogset, I couldn't screw on the lockring that came with it. I had to use the lockring from the old cogset.
how old was the old cog set? your chain might have worn into the pattern of your old cog set causeing skip shifting, My advice would be a new chain and then if that doesnt work new chain rings.
Try cleaning and lubing the shifter cable or replacing it with a new one. When cleaning the old one check for broken strands which may causing the cable to hang up instead of moving freely.
Whenever I have shifting problems that I can't fix by turning the barrel adjuster, I go back to square-one and "re-install" the cable, before replacing parts helter-skelter.
Sounds like excessive cable-friction and/or an off-by-one-click alignment issue.