Sy Reene beat me to it. I'd start with verifying the hanger's aligned.
There are two very different issues being discussed in this thread.
Skipping, i.e., the chain staying inline but riding over teeth on the cog(s). This happens when the cassette and chain are no longer engaging perfectly. Cause: the cassette is worn, the chain is worn, a.k.a. stretched, or both. Oh, this can happen with chainrings... that's just pretty rare.
Chattering, jumping from one cog to another. This has lots of causes. But chain and cassette wear are usually not the cause.
- Limit screws are usually not the cause either. That's a "set and forget" thing. Assuming they were adjusted correctly when the derailleur was installed, they can usually be left alone.
- Cable tension - quick and easy. Cables used to stretch more than they do today. But still - putting a twist in the barrel is an easy solution... sometimes.
- Cable and housing condition; when did you last re-cable? Dirty, corroded (sweat, etc) cables can add drag and fight with the derailleur's spring. A kink or very tight bend can add drag too - but you'd have seen that by now.
- Derailleur hanger alignment. It doesn't have to be out much. If you find that the derailleur is good on some cogs, not in others and adjusting the cable tension just moves that chattering around to adjacent cogs - that's a red flag for the derailleur alignment being off. If you don't have an alignment tool, have the shop do it. And yeah... they should have checked that already.