The basics.
Trigger control, sight alignment, sight picture, good stock weld and consistent eye relief. Slow, steady squeeze with consistent follow through on every shot. Don't expect the shot, let the rifle surprise you every time. I'm sure you know them and I'm simply listing them for everyone's benefit - certainly not to be condescending.
No matter the rifle, as long as it's not a total piece of trash then the application of the basics will bring results. I can only imagine what the right hand/left eye dominant thing does to complicate the application of the basics. I would recommend switching to the left hand. The initial change over may be daunting but I really believe in the long run you'll get the results you're looking for by making the switch.
The basics are require but if you shoot cross eye then you put side load/strain on the rifle. The only pressure on the grip and trigger needs to be straight to the rear and shooting cross eye it is almost impossible to have only rearward pressure.