The LimbSaver/"Sissypad" will probably lengthen the stock a bit, which will help you. I like using one that velcros onto the butt.
https://www.midwayusa.com/product/1006452941?pid=547301
I have also used pads that strap on me:
https://www.midwayusa.com/product/1002530831?pid=680235
The typical US-made commercial shotgun has a "Pull" of 14-and-a-quarter inches, which is probably going to be way short for you. The Galco pad which I linked to above would add about 3/8", which may be about right for someone about 6 feet tall, but certain habits cause variation. One habit I picked up from rifle shooting which causes me to need much longer stocks is always running my face way forward on the stock. You can experiment with even more length, putting resilient spacers in-between the buttplate and the Sissypad. I'm 6'4", and I need a 16" pull.
My daughter is cross-dominant, like you, right-handed and left-eyed. I have encouraged her to shoot left-handed. If you shoot right handed, you will have to close your left eye, reducing situational awareness. I have weak right eye dominance, which will show up when I am really tired.
You would do well to shoot it enough such that reloading is not something you consciously think about. Long ago, the people that taught me to shoot Skeet told me they could shoot faster with a pump than an auto.
Better fit will reduce the sense of recoil. The sissypad will help also. Shooting lighter shot charges will also help. For a long time Aguila loaded these cute little mini-shells which typically have 5/8 oz. shot charges, or slugs in the same range. I can shoot 4 rounds of 12 gauge Skeet with such ammo before I get tired, but with ounce-and-an-eighth I get tired after 2 rounds. An additional benefit with them is that some shotguns will function reliably with the mini-shells, and your magazine can hold a lot more of them than 2-and-3/4 inch shells. They did not work reliably through the magazine of my 500. They shoot single loading in a 500 just dandy. I usually shoot them in a Double.
https://www.midwayusa.com/product/2506255579
https://www.midwayusa.com/product/2506419359