Should I switch to a 13-23T or 12-25T cassette?
Should work. Put a new chain on.   

New chain as a worn chain will wear the cassette as it is stretched and does not match up.
If the chain is severely worn it may even jump.   

Chain is cheaper and should be replaced unless it is almost new.
A cassette will typically last 2-3 chains.  

If you sometimes use 28 then why are you giving that up?
Is a closer gearing really worth giving up the range?
Why not like a 12-27T?