Check to be sure the gun is fitted to your hand properly. A grip that is too fat for your hand can really mess up your ability to grab the pistol in a consistent grip every time and you find that you're readjusting every tiem you draw. If your natural finger reach doesn't put your finger in the right position (see below), you may need a different grip or different pistol altogether. All else being equal (and correct), this by itself could solve your problem.
As you get a grip on the pistol, make sure your grip is tight and firm but not so tight your muscles start to shake. The web of your hand between the thumb and forefinger should get squished up a bit.
Bury that trigger finger into the trigger guard so that the trigger sits behind the first knuckle, not on the fingertip. It's not how you've been taught for target practice but it works very well for SD and SD practice. The tight grip gives you the ability to grab the pistol more consistently every time and reduces the possibility that you'll lose the pistol to an attacker.
Draw up to your chest or even as high as your shoulder (same side of course) and then push the gun out toward your target. Raise it as high as is comfortable for you. While doing so, make sure your other hand comes to the grip as well for a two-handed hold OR be sure you get your other elbow out of the way. You don't want to shoot your own arm in the process!
(Raise your other elbow up past your eyes with your forearm resting on your forehead, palm out, while the gun is still at your chest level. It can be an additional defensive maneuver in close-quarters combat and can prevent your gun from being taken from you.)
All of the above can be adjusted to your needs and ability. Even if you don't do everything as described (hard to describe in writing anyway!
) it can give you some ideas about how to adjust your draw and presentation to the target to "fix" your problem.
Always practice slow. You HAVE to develop muscle memory and if you practice fast and get it wrong half the time, you're not building that muscle memory. Better to practice a thousand times slow and get it right EVERY time than to practice one time fast and get it wrong. I had an instructor years ago who said, "For every one mistake you make, you have to get it right 100 times to get rid of that mistake." (BTW, that's true of basketball, martial arts, guitar. . . . Whatever you're learning.)
Let us know how it goes.