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.

Hillfolk/Dramasystem and Cortex in it's Drama implementation are the best I know of for heavy social games.

Fate uses the same rules for social conflict as physical conflict. You (probably) use different skills and stunts, and (probably) target a different resource, but the surrounding rules are the same.

If you want to stab someone to kill them dead, you'd roll something like Swordfighting vs Athletics. If you roll better, you inflict Physical Stress on them. If they don't have room to hold the stress, they can convert some of it to a Consequence instead. That might be something like "broken arm", "ringing ears", "bleeding a lot oh god", or what have you. If there's still stress left over, they're taken out of the scene. Maybe you run them through. Maybe they get taken prisoner. It's up to you. You won the conflict.

If you want to insult someone to make them cry, you'd roll something like Provoke vs Will. If you roll better, you inflict Mental stress on them. If they don't have room to hold the stress, they can convert some of it to a Consequence instead. That might be something like "Humiliated", "Obsessively thinking of what I should have said instead", "Crippling Self Doubt", or what have you. If there's still stress left over, they're taken out of the scene. Maybe they leave the room sobbing. Maybe they just give up. It's up to you. You won the conflict.

It's a good system, but I have found it requires players to be a little more engaged than "i hit him with my sword" or "can i roll insight?" like you sometimes get with D&D. I like that aspects (like the consequences I gave as examples) are made up on the spot. So long as the table agrees it fits, you just jot it down and move on. You don't have to deal with a long list of pre-written things. That is probably a factor in why it's harder to phone in like D&D.