When I'm unfit, my stronger arm gets more pumped than my weaker one. I don't notice the difference when fit though.
If you're shoulder isn't causing you any trouble, and you're aware that you have bad posture etc, then why go see someone?
I also notice such imbalances, and track them down to too much right-handed during puberty.
fiddle with gear with the dominant hand
When I was a traddie I had better forearm endurance in my non-dominant arm because of the natural tendency to fiddle with gear with the dominant hand if possible, thus spending more time hanging on with the other hand.