Originally Posted by FBinNY

A few possibilities come to mind.

The first is that the chain and freewheel aren't ideally matched for clean shifting. Index shifting requires good matching because down shifting counts on the chain's outer plate snagging a tooth and beginning to climb up without much overshifting. By the same token, upshifting requires the chain to release and drop without overshifting in that direction.
So there's a possibility that your specific combination of chain and freewheel is working against you a bit. OTOH, they often settle in and shift better after some miles.

Before spending dough here are a few other, easier to play with things to try.

1- B-screw adjustment. The closer to the sporcket the jockey wheel is, the crisper the shifting, until you get to close and shifting tends to jam. Try bringing the RD closer via the B-screw (take weight off the screw by pulling the RD back) while you adjust it). Get it as close as possible without coming closer than 5mm or so on the larger sprockets. If this helps a bit, go closer,but never so close that the pulley, chain and sprocket all touch.

2- cable friction. This only affects upshifts, but often people need to bias outward to compensate then have poor downshifts. You can test whether cable friction is an issue, by adjusting the trim based on good down shifts, then upshifting one click, and if/when the RD hangs, plucking the wire like a guitar string. If plucking completes the shift, cable friction is the issue to be corrected.

3- floating upper pulley. Most (all?) Shimano RDs have a floating upper pulley, that can move side to side a few millimeters. This makes trimming easier because the float allows the pulley to center itself. OTOH that loat absorbs travel when you shift and can cause sluggish shifting. Try switching then upper and lower pulleys and see if that improves the shifting. If so. you might find trim a bit difficult to maintain and have a noisier drivetrain. Whau I sometimes do, is use an O-ring, or DIY spacer on the upper pulley bolt to reduce but not eliminate the float. This is a tials and error process looking for the best balance between crisp shifting and forgiving trim.

So, you now have some things to try, none of which cost anything but time. If these don't resolve the issue, adjust as best you can (I prefer an outward v=bias, bcause I can easily shift beyond the click a half space and let the lever settle back after shifting). Then cross your fingers hoping that some break in solves everything.