I assume you hold your shotgun and rifle to your left shoulder?
Thats how I teach it anyway. for ex, right handed, left eye dominant
handgun - hold gun in right hand, line up with left eye, both eyes open
shotgun - hold gun to left shoulder, sight with left eye down the bbl, both eyes open.
Rifle, this is more of a tossup. For what sounds like the short range shooting you mentioned, I'd think holding to the left shoulder would work best, since it would facilitate instinctive point shooting.
long range work with one eye closed, Its kindof a tossup. To you want a steeper learning curve, then shoot right handed. Do you want to be able to possibly shoot with youer weak eye open all the time? Then go Left handed.
I'm a thoroughly mediocre shooter, but I am VERY left eye dominant. I find I can shoot a scoped rifle with both eyes open and it doesn't bother me.
Re the topic of figuring out what eye is your dominant one. The old method of focusing on an item then pointing at it with your finger and seeing which eye it lines up with works best when the subject does NOT know what you are doing. That way they can't let their preconceived notions guide them.
I hope this helps some bit.
Don