TargetTalk

I am a left handed shooter at the college level and am having problems in air rifle standing.
In Air rifle my hold up and down is very consistent but side to side I see what appears to be a distinct pulse from my chest transferred to the rifle, making my hold quick and hard to predict making nines on either side of the 10 almost constant. I have been shooting low 570's.
I have tried adjusting NPA and moving the butt plate further out on my arm, and also have tried "chicken winging" it with my left arm, "mind you, I am a lefty" in order to reduce the pressure on the rifle from my chest without much result. The other problem I have is when I reduce pressure on the buttplate the rifle wants to fall forward out of my shoulder, and I feel like I am pulling the rifle back into my shoulder. Also I am fairly tall also but I feel that my body's horizontal movement in position is under control and thus not the source of the problem I am seeing.

Sorry for the long post, any help or suggestions would be greatly appreciated, especially from left handed shooters!
Hopefully I am not crazy and some other left handed shooters have noticed this also.
Styles

Surely this has nothing to do with being a lefty? Is the situation not completely symmetrical?

Are you using a palm rest? If so. you also need a butt hook. That will stop the forward drop of the muzzle which you seem to suffer from. If butt hooks are illegal in your discipline, you have to adjust the forend hand support or palm rest attachment point to minimise the tendency for the muzzle to drop. Or add counter weights aft of the forend support point, which introduces other problems!

Well for a lefty there is a stronger pulse from the heart in the chest area. I am using a butthook in smallbore but sometimes fear how if I let the rifle pull forward until the hook stops it, then it creates less contact with my arm over the rest of the butt plate creating more problems and more left right sway.
I added weight to the back of the air rifle and that seemed to help it wanting to pull out of my shoulder. but the smallbore 1813 is much harder to add weight onto and its heavy to begin with