Here's the situation:

When you have a weapon in your right hand, blessed ring of damage gives +7 to damage (possibly it's variable but in my current game, the damage goes from 5d6+20 to 5d6+27, I'm using executor).
When you put the ring on your left hand, the bonus is ~halved (+3).

However, I would expect that if I wield a weapon in my left hand and equip the ring of damage on that hand as well, the damage would again be +7.
It's still only +3.
I believe it's hardcoded that RoD grants higher bonus on the right hand and lower on the left, no matter which hand actually wields the weapon.

Instead, if RoD is on the same hand as the weapon, the bonus should be +7, no matter if it's left or right.
The chance to hit and damage are identical, whether you're right or left handed, thus I don't think adom has any concept of a dominant hand, with the exception of rolf's axe+shield combination.