I'm cross dominant (right handed, left eye dominant) and shoot left handed for rifle, shotgun, and bows (compound o recurve). This is 100% the way to go. All the little tricks to switch or fool your natural eye dominance are a psychic drain and rob from mental energy needed to execute shots. Some will disagree.
I've done a lot of coaching for shotgun and archery at work and seen people at these classes (and buddies in the hunting woods) either succeed or struggle based on this choice. When a shooter is addled in the brain while trying to crush a moving target, you can tell and they miss a lot.
The actual gun doesn't matter as much. Shoot what you can afford/have. Gun fit is important, and a lot of newer guns allow for adjustments to cast, drop, etc. using plastic shims.
I shot right-handed rifles for years, but eventually switched to a LH Tikka this year and am liking it. I shoot right-handed auto loaders with the stock shims in place to help my left eye line up with the barrel. I shoot left-hand over unders.
I don't tolerate ill-fitting guns and want to flow with my shots without thinking. This is what's worked the best.