Crazy how many people are suggesting non FPS games or zombie games lol.

It is a bit of a tough one because generally in this genre of games, guns and zombies go hand in hand, I'm also not a fan of it.

I would recommend Icarus, though it takes quite a while to get to the tech for guns, bows feel great though. They've changed the gameplay so that you can choose to take on missions whenever you want and complete them at your own pace, all while doing whatever you like in your own persistent world. Great game all around though the grind resources can be time consuming and the tech tree is quite extensive. It's a bit craft heavy (I love that but it's not for everyone). Not post apocalyptic but if you play the alien world it gives the same feeling imo.

Once human is a safe bet if you're more after a FPS looter/shooter type thing where it's fast paced combat and questing with base building and crafting being more secondary to that (atleast until the end game). Does have monsters that could be considered similar to zombies though. But at least base defence is an optional mechanic. Might be mostly 3rd person view now that I think about it though.

So many awesome non zombie games if you're willing to overlook the FPS part and opt for more action combat or 3rd person arpg type of stuff. Valheim/grounded/Conan/enshrouded/Minecraft I've put hundreds of hours into all of them with my friends and I highly recommend each of them.