It really depends on what you want to emphasize in your game. The Prism Pentad novel series starts with slaves toppling a corrupt king and a fight against evil, and goes from there. Dark Sun starts as a grim setting, but the metaplot turned that on its head very quickly, with mixed approval. It’s an example of a heroic sword and sorcery game.
Post Apocalyptic is more atmosphere than anything else. You can run all of the same stories in a post apocalyptic game in any other setting, with minor adjustments.
You can pick one and run with it. But the political climate in Dark Sun is stable - there’s no reason why it has to shift in any dramatic way. In a gritter game, nothing changes, or the changes aren’t nearly so heroic. Killing Kalak and having Tyr as a “free” city is perfectly fine in Dark Sun, if its toned down.
A few individuals find that attaining power isn’t so easy, and even if they do, they can’t get everything they want. Some reforms on slave treatment, some changes, but in the end, there are still slaves, life is still harsh, and the new regime has to make many compromises to maintain stability. It can work even better in this light if Kalak is not a Champion, and kept the heart of a slain Champion, gifted by Rajaat. That way, the city can keep its Templar, and the realignment will be heavily political, with lower ranked templar getting promotions as old ones are removed, but the mechanism of power demanding concessions to keep them loyal. More traditional sword and sorcery in theme, if not power level.
Or, you can focus on the villages and wastes. This seems better for Sword and Planet. The city-states are the hubs of civilization because they have the resources, the people, and offer advantages and safety that the wastes don’t offer. The price is steep, but many prefer to live in fear than die free. It’s dangers aren’t the dangers of a dying planet, pressing down on the character, but a fight with others over who gets the guarantee of the future a city offers. Even if there’s a famine, the city is likely still the best off. In the wastes, your next meal isn’t in sight, or a handful of cp away. You struggle for every drop of water. Villages have more, but raiders are a threat, and there are still no guarantees. Defilers are key here. The PCs find a small oasis? They come back one day, and it’s reduced to ash. A village grows to large? It’s harder to defend, and the Sorcerer Kings may come. Druids, and perhaps the Veiled Alliance offers some help, if the land is respected, but the forces who will exploit that land are many. The land itself is the treasure, and the characters are going to have barter part of it get the allies they need to defend the rest.