You need a good instructor that can watch what you are doing as you shoot. Find a local NRA Bullseye club. Watch. Learn. Sounds like you are convulsing your whole gun hand, looking OVER the front sight to look at where the rounds are going - not watching your front sight, ignoring the "wavering", just maintaining sight alignment while pressing the trigger straight back until the pistol surprises you and discharges; then following through while watching the Front Sight ALL the way through recoil and settling back into the notch of the rear sight, and reacquiring the proper sight picture. Do not look at the target to see where the bullet went right after the shot until you have followed through - reacquired the sight picture and have recalled what the sight picture looked like when the pistol went off!
Get Gil Hebard's book. Read it, then get a copy of The Modern Technique of the Pistol. Read it. Then look for a Randy Cain Handgun 101 class.