I actually went the opposite direction. I added "handedness" back, so you don't get your proficiency bonus using your off-hand. Otherwise no "I am not left-handed!"
If you take Dual Wielder you are proficient with both, and receive your normal damage bonuses.
In addition, the AC bonus is +1 with two weapons, or +2 if using a weapon and a buckler or a weapon and a cloak. You can still make an off-hand (punching) attack. The buckler can be spiked to cause more damage, and the cloak can be used as a special attack to attempt to restrain or disarm the target with proficiency. In my campaign, special attacks like this grant the target a saving throw (DC 8+ your attack modifier), and if the attack fails they get an opportunity attack against you.
The idea was more for flavor, and "realism". I don't like that you can't model Inigo Montoya or Jamie Lannister with the current rules. I also encourage other options than lethal attacks, and one of the main purposes of dual wielding is defensive.
I disagree that Great Weapon Master is much better. There are significant penalties to get that extra damage. It works out to about 4 hp/rd for Dual Wielder (using 2 1d8 weapons), or 5.5 hp/rd for GWM unless they score a critical (and then hit with their bonus attack. But assuming a 50% chance to hit, the dual wielder will hit once every round on average, where the GWM using his maximum penalty/damage bonus will only hit once/4 rounds. Big hits, but at a price, particularly since combats can be lethal quickly.
Ilbranteloth