First, a note on Social Combat. While Exalted's system has many flaws (including allowing you to stick your fingers in your ears and not listen), there are still hints there that can help you with your system.
First, social conflict is aggressive and proactive. The participants must have goals.
Second, the system must encourage small changes. As you note, D&D's one roll method for Diplomacy is dull. Exalted's Social Combat covers this with Intimacies. Forcing participants to slowly adjust another character's Intimacies over time to persuade them creates opportunities for thrust and counter-thrust.
FATE Core is an excellent system that takes a step back from the granular details that games like D&D and Exalted focus on. It unifies conflict under a simple model that can be used to represent all kinds of conflict, including economic, social, physical and more.
A conflict consists of using skills to Create Advantage, Harm and Defend. The goal is inflict enough stress (aka damage) on your opponent that they opt to concede (and they will have some control over the result of the conflict) or Take Them Out (so you have full control over the results). During the conflict itself, you can protect yourself from being taken out by accepting semi-permanent consequences that are related to the current situation.
This means that you can play out a scene where you and other characters who disagree on something can manuever around each other and hash out your agreements while also possibly taking some fallout from it.