So, I'm not a huge fan of two weapon fighting fighting style or the Dual Wielder feat. Cinematically, they just don't make a lot of sense to me and are pretty darn ineffective means for combat. Characters that would wield a weapon in each hand ought to strike some kind of balance and the rules as written seem clunky. With that in mind, I'm playing with a homebrew feat.
Off-Hand Weapon Fighting
You can use two-weapon fighting when your off-hand weapon is light even if your main-hand weapon is not light. You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand if one is light.
You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.
As a bonus action you may attack with your off-hand weapon and add your ability modifier to the damage. If the damage die is less that 1d6, treat as 1d6.
As a bonus action you may take a defensive stance with your off-hand weapon to take the Dodge action.
If you are able to take two bonus actions on a turn, you may not take both of these bonus actions.
The first two items are modified right from Dual Wielder. I wonder if replacing the "Dodge as bonus action" with "gain a bonus to your AC equal to your proficiency modifier" (a la Defensive Duelist) would be more reasonable, but I have come to love Dodge. Also, the +1 AC from Dual Wielder (which I've included) seems a little strong, especially early in the game, and perhaps should be replaced with +1 Dexterity/Strength.
I very much like the idea of the swashbuckler who parries with the dagger and many classes, especially Fighter and Ranger, don't have a ton of bonus action abilities, so offering two that don't explicitly add damage (a la Sharpshooter) feels fun. It also reflects some of the fighting in the Witcher series between power and defensive style fighting.