By book rules, there's no dominant hand distinction, or really 'off hand' distinction at all - just rules that say things like, "When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand."
They generally avoid talking about 'off-hand' if they can do so without being confusing.
As for the game, you currently cannot wield your shield in your right hand and weapon in the left. It won't allow it.
I'd really like to be able to set a martial dominance, to flip the stances that the models use left or right.