I doubt it's the shifter, but rather the less expensive issue of cable friction because of dirt or a worn liner, a trim adjustment, a bent hanger, or the "B" screw setting.

Do a quick test of the RDs shift response shifting by pulling the bare wire away from the down tube like a bow string. It should be smooth and free as you run it up and down through the gears. If the RD motion is at all jerky, the rear housing loop is the likely culprit.

If the action is smooth shifting from the downtube, try adjusting the cable a bit tighter. This should improve the shift to larger sprockets though at the expense of more sluggish shifting in the other direction. If you can get all the shifts up to larger sprockets to work well but it hangs on the return, try plucking the cable at the downtube. That pulls down a bit of cable that may be hung up in the HB to frame housing section. If Plucking works, you need to lube or replace the housing.

Also check the height of the RD on each sprocket. There should always be daylight between the upper pulley and sprocket with about 1" of chain free between them. Watch as you shift to the next geat larger, if the RD seems to stutter or get pushed back, it's too high. Tighten the "B" screw to push it lower and things should improve.

All the above assumes a straight hanger. If you can't spot a cause of your headaches, have this checked and eliminated before wasting any more time.