There are several methods for left hand cross drawing out of a holster when the right hand is injured. The best I have ever been exposed to was given to me by Bill Bellman. For many shooters this will be a new skill so do not strive for perfection. Work for safety then competence. Speed and fluidity will come with proper practice and repetitions. Please remember to begin your cross-body left hand snub drawing skills with an unloaded weapon and keep the muzzle under control and pointed in a safe direction.
Starting with the snub in a strong side holster reach across with your left hand. Seize the snub around the stocks with your left hand’s thumb pointing to the rear and your palm turned in toward your ribs.
Unsnap any safety straps and lift the snub up and clear of the holster. While maintaining a good grip on the stocks, bring the snub around to your weak side. Keep the muzzle pointed down and make no effort to adjust the orientation of the snub untill you have moved the snub to the left side of your body. Idealy the snub will end up near your left hip. Your left hand thumb and the butt of the snub should now both be facing forward with your left hand at or near near your left hip. Being careful not to make contact with the trigger, stuff the nose of the barrel into your waistband. Do not let go of the snub as there is too little barrrel length to keep the snub in place. Using your belt line as a temporary shelf only and without losing control of the snub, rotate your left hand 180 degrees. You should end up with your knuckles turned into your ribs and your thumb pointing rearward. Re-grab the snub. Your new grip on the snub should resemble a classic cavalry draw. Lift the muzzle up from behind the belt line only high enough to clear the top of your waistband. Once you have cleared the top of the waistband lower your hand to full arm extension. Then rotate the gun 180 degrees before lifting the snub to firing position.
Note that the usual method to complete a cavalry draw after your hand has ended up at the knuckles in, thumb pointed to the rear position is to lift your hand as high as possible then rotate the weapon 180-degrees while keeping the muzzle pointed down. While the traditional Calvary style cross draw is reasonably safe with a long barreled weapon often the shooter trying this with a short barreled snub will rush the job and inadvertently swing the muzzle across his own ribs. The Bellman’s method of lowering the snub before rotating the gun 180-degrees removes this risk