Hillfolk/Dramasystem and Cortex in it's Drama implementation are the best I know of for heavy social games.
What Should I Play?
Some quick recommendations, let me know if you want to know more about any
Genesys - Uses a separate wound tracker (Strain) for all kinds of mental fatigue. Including its "debate" system for when a single roll isn't enough. Essentially it works like a combat with turns and all. Its narrative dice system is good at keeping things non-binary so it does drive the narrative by itself. There is also talent support for those who want to build for social encounters.
Ironsworn - Essentially PbtA at the core with basic social moves. When more granularity is desired the debate can use the fight framework. Why I mention it specifically is also its really awesome approach to trackers. You don't have to fill the tracker but can roll against it whenever you feel the goal is met.
Burning Wheel (and Torchbearer) - Uses a rock-paper-sissors like system where each side's action is played against the other's. Really cool system.
Exalted 3e - Have the least experience with this but the one that I like the most of the meaty systems. It boils down to manipulation of characters Intimacies. Everyone has them. They are things like Love (person) or Affiliation (group). And you cannot convince anyone without tagging in one or more Intimacies, unless you threaten them. The more socially sly a character is the harder it is to find out about their Intimacies so there is a whole system for that too. So you may start engaging with someone to first discover an intimacy or two that you then can use to manipulate them. Once manipulated to a suitable level you can "cash out". I may have some details wrong, was a while. If interested I'll pull out the book and freshen it up.
I second Genesys especially for social encounters. It doesn’t over complicate but enables doing “damage” in a back and forth between sides through opposing checks depending on the argument approach. Also as a system it greatly lends itself to role playing and descriptive play through the mechanics themselves.