no shortcuts
Unfortunately it's going to take practice.
With enough rounds downrange, and for close targets (20 meters) you'll eventually be shooting your handgun without USING the sights at all. This skill (instinctive shooting) can be developed using 8 to 10 steel targets (each about 4" diameter) in random patterns, and at different distances (from 7 to 20 meters out). The satisfying report of the round impacting steel will indicate a hit. This type of setup on a range will allow you to transition from sights (fundamental marksmanship) to instinctive-level proficiency. Steel targets also eliminate the need for changing paper, and they can be rolled around the range to accommodate a variety of configurations.
Eventually, you'll be using both eyes wide open, and looking at your target continuously rather than your sights. Your eye will still be looking directly over your sights, but your focus will be on the center of the target. If you're doing it right, you'll be scanning, plinking, scanning-plinking, etc. until you've nailed them all.
Also, I wouldn't expect to employ this skill at targets any further than 20 meters max. This is a skill designed to teach you how to engage multiple targets smoothly, and you can incorporate other drills with it, including failure to fire/feed, movement, cover and concealment, etc.