I would try cleaning and lubing it all first and also replacing the gear cables before replacing the cassette. It sounds to me like the gear cable may be sticking and preventing it shifting cleanly to smaller cogs. If the cassette is badly worn, then the chain and chainrings probably will be too, so you will need to replace all of them at the same time.
If you’re only using the small ring on a double ring setup, the chainline will probably be horrible in the small cogs so that will cause premature wear and possibly poor shifting. If you want to convert to a single ring setup, you’ll need to make sure that your chainline is ok. Personally, I would stick with the double setup and save the small ring just for steep climbs, with the big ring used most of the time (how they’re designed to be used).